ARMY RESEARCH LABORATORY 

Natural Computing: Analysis of 

Tables for Computer 

Representation 

Approved for public release; distribution unlimited. 

20000310 092 ~ 
The findings in this report are not to be construed as an 

official Department of the Army position unless so 

designated by other authorized documents. 

Citation of manufacturer’s or trade names does not 

constitute an official endorsement or approval of the use 

thereof. 

Destroy this report when it is no longer needed. Do not 

return it to the originator. 
Army Research Laboratory 

Adelphi, MD 20783-l 197 

ARL-TR-2041 February 2000 

Natural Computing: Analysis 

of Tables for Computer 

Representation 

Som Karamchetty 

Information Science and Technology Directorate 

Approved for public release; distribution unlimited. 
Abstract 

Some fundamental objects in practical documents have not been 

implemented in software so that they can be used easily for 

calculating. One such object is the table-despite the mistaken view 

that databases are adequate representations of tables. A survey of 

practical tables found in a variety of real-world documents reveals 

that many of their useful features are not captured in software. This 

report proposes data structures and computer representation for 

table objects. Through the adoption of such structures and 

representations, practical table objects can be developed for use by 

domain specialists. Such tables embedded in electronic documents 

can be used in interactive applications to retrieve data, but most 

importantly, they can be used as functional representations for 

copying and pasting into procedures and programs. Use of these 

table objects, together with other natural computing objects (such as 

equations, graphs, and procedures), will permit electronic documents 

like handbooks, textbooks, journals, and bulletins to be used 

seamlessly for calculations by both domain specialists and naive 

users. Such developments will reduce the lag between the 

availability of information and its use in calculations, and encourage 

the further development of knowledge. Software development for 

computation will change and its costs will be contained. 
1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

Contents 

Introduction ................................................................................................................................... 1 

A Textbook Example of Calculation Features .......................................................................... 2 

Usage of Tables .............................................................................................................................. 6 

Ends and Means ........................................................................................................................... 12 

Treatment of Tables in Text Formatting Tools ........................................................................ 13 

Informal Survey and Analysis of Practical Tables ................................................................ 14 

Anatomy and Morphology of Tables ....................................................................................... 27 

7.1 Cell Cages ................................................................................................................................ 28 

7.2 Cells ......................................................................................................................................... 32 

7.3 Table Types ............................................................................................................................... 32 

Computer Representation of Tables ......................................................................................... 34 

Choice and Use of Tables .......................................................................................................... 

.37 

Search for Data ............................................................................................................................. 38 

Visibility of Table Data ............................................................................................................... 39 

Testing a Table in Isolation ........................................................................................................ 40 

Embedding Tables into Text ...................................................................................................... 41 

Tables and Databases .................................................................................................................. 43 

Example Usage of Natural Computing Tables ....................................................................... 46 

Conclusions .................................................................................................................................. 51 

Acknowledgments ................................................................................................................................ 52 

References .............................................................................................................................................. 53 

Distribution ........................................................................................................................................... 55 

Report Documentation Page ................................................................................................................ 57 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

10. 

Figures 

A sample page containing text, sketch, and equations ............................................................... 3 

A sample page containing text, graph, and equations ................................................................ 4 

A sample page containing text, table, and equations .................................................................. 5 

Status of baseball games on a given day shown by a number of tables ................................... 6 

A sample table showing properties of saturated steam ............................................................ 10 

Example of a simple table with its parts identified ................................................................... 14 

A sample table with a variety of data types ................................................................................ 15 

A sample table showing distinct data types ............................................................................... 15 

Sample table showing algebraic expressions .............................................................................. 16 

Example of complex case and switch statements required to replace a table such as 

that in 9 ............................................................................................................................................. 16 

... 

111 
11. A sample table showing a conditional range value in an element .......................................... 17 

12. Example table where elements develop branches as they progress along columns ............. 18 

13. A number of columns develop variable numbers of branches or multiples.. ........................ 19 

14. A sample table with both a column header and a row header ................................................ 20 

15. A sample table showing ranges with value correspondence ................................................... 21 

16. A sample table connecting values to dimensions in a sketch.. ................................................. 22 

17. A sample table showing high-density polyethylene tank data having three size values 

in one column .................................................................................................................................. 23 

18. A sample table containing pictures and collapsing branched rows ........................................ 24 

19. A sample table containing graphical icons ................................................................................. 25 

20. Sample tables where elements can contain values of computations.. ..................................... 26 

21. A generic structure for a simple table .......................................................................................... 27 

22. Cell cage and cells ............................................................................................................................ 28 

23. Cell-cage sequence numbers ......................................................................................................... 29 

24. An example table showing a column with branches ................................................................. 30 

25. Representation of column header cell cage for table in 24 ....................................................... 31 

26. Cell-cage sequence numbers for a column header with branches .......................................... 31 

27. Example cell types with sample data .......................................................................................... 32 

28. A spectrum of tables in profile representation that can be generated .................................... 33 

29. Iterative cycle of table development in future Natural Computing system .......................... 33 

30. A hierarchy of table classes and component classes .................................................................. 34 

31. Development of a superset table from two tables ..................................................................... 36 

32. Development of mathematical operations with tables ............................................................. 36 

33. Adjacency relationships are used to get table values ................................................................ 38 

34. Example of a table embedded in text ........................................................................................... 42 

35. A table of baseball results representing a player and his position in same element ............. 45 

36. A table embedded in text with Natural Computing menus.. ................................................... 47 

37. A table activated for interactive use ............................................................................................. 48 

38. A table being set into a procedure ................................................................................................ 49 

39. A Natural Computing screen showing a document with a procedure.. ................................. 50 

Tables 

1. Example showing two columns of data that hold a discrete relationship ............................... 7 

2. Example showing two columns of data that hold a continuous relationship ......................... 7 

3. Sample of logarithms ....................................................................................................................... 9 

4. Tabular representation of temperature-volume-pressure relationship for an ideal gas.. ..... 

.ll 

5. A in a word processor showing an element spanning two columns ...................................... 13 

6. Example of a with a forbidden repeating group ........................................................................ 44 

7. Example of a with a forbidden true hole.. ................................................................................... 44 

8. Example of a containing forbidden internal structure .............................................................. 44 

9. Example of a with an allowable structure to represent information in 8 ............................... 44 

iv 
1. Introduction 

In earlier work (Karamchetty, 1997 and 2000), I have described an approach to software for computation that is based on the way people 

actually compute, using available information in a variety of formats. 

This report builds on that discussion by examining one aspect of Natural 

Computing: the use of tables. Before the invention of computers, most 

knowledge was captured in the form of books and other paper documents. Books can be further classified as textbooks, reference books, 

handbooks, and journals, based on the temporal nature of the information. More transient documents are flyers, brochures, and receipts. Information was printed on paper for storage, retrieval, and communication. 

The paper-based information was read by the end user. By reading the 

information from one or more documents and by combining it with one’s 

own intuition, invention, and discovery, one generated and printed new 

information in the form of another paper document. 

In the paper-based universe, when people dealt with technical matter, 

domain knowledge was captured in the form of text containing equations, tables, graphs, and pictures. Without pictures, descriptions of 

scenes were elaborate; as we recall, “a picture is worth a thousand 

words.” In technical subjects, the pictures could be sketches, schematics, 

drawings, paintings, or photographs. Sketches stood for descriptions of 

parts and components in terms of shapes and sizes. Schematics and other 

diagrams showed the mutual relationships of components in a system 

and the state of the system and its temporal nature. 

Both tables and graphs captured relationships among sets of variables. 

Graphs additionally provided a highly visual insight into the mutual 

dependency of the variables. Domain specialists read the text and concurrently used the included tables, graphs, and charts. They used note pads 

to make temporary notes and calculations. Simple calculations were done 

mentally. More complicated calculations required aids, such as log tables 

and slide rules. Domain specialists captured new ideas and information 

in the form of more equations, tables, graphs, and pictures; appended 

them to text; and communicated the new documents to others in the field. 

Documents were subject to three principal types of use: (1) reading and 

comprehension; (2) interactive calculations using the tables, equations, 

graphs, and pictures along with the text; and (3) development and recording of new functions (tables, equations, graphs, and pictures). Capturing 

these essential natural forms and processes in a computer software 

system is the goal of Natural Computing. 
2. A Textbook Example of Calculation Features 

Figure 1 shows a sample page from an engineering textbook describing 

mechanical springs. The page consists of a sketch of a mechanical spring, 

text, and equations. Figure 2 shows another sample page with a graph, 

more equations, and more text. By reading the explanations on these 

pages, an engineer can understand the domain of mechanical springs. By 

studying the graph on the page, the engineer can understand the trends. 

At any time, the engineer can obtain values given by the graphs-this 

activity is usually called reading a value from a graph. While using these 

pages, the engineer starts with values of D and d, proceeds to calculate 

the value of the variable C from equation 8-l in figure 1, and reads the 

value of the Wahl correction factor K from the graph of spring index 

versus stress correction factor. This value of K and an input value of F 

(force) are next substituted into equation 8-4 (fig. 2) and the value of stress 

(2) is calculated. The engineer may next proceed to the sample page 

shown in figure 3 and read the values of A and nz for a given material. 

These values are substituted into equation 8-10 (fig. 3) for calculating the 

ultimate strength in tension of the spring material. 

This description shows how domain specialists present information in 

textbooks for use by readers in performing calculations. Thus, textbooks 

are used both to explain the subject and to provide information in the 

form of text, sketches, equations, graphs, and tables for ready use in 

calculations. 
(b) 

FIGURE 8-l 

(a) Axially loaded helical spring; (b) free-body diagram showing that the wire is 

subjected to a direct shear and a torsional shear. 

the hose in a straight line perpendicular to the plane of the coil. As each turn of 

hose is pulled off the co& the hose twists or turns about its own axis. The flexing 

of a helical spring creates a torsion in the wire in a similar manner. 

Using superposition, the maximum stress in the wire may be 

using the equation 

,Tr F 

7 max =kJ+- A 

computed 

(a) 

where the term Tr/J is the torsion formula of Chap. 2. Replacing the terms by 

T = FD/2, r = d/2, J = nd4/32, and A = 7rd2/4 gives 

8FD 4F 

t= -+z rd3 (6) 

In this equation the subscript indicating maximum shear stress has been omitted 

as unnecessary. The positive signs of Eq. (a) have been retained, and hence Eq. (b) 

gives the shear stress at the inside fiber of the spring. 

Now define spring index 

D 

c=J (8-l) 

as a measure of coil curvature. With this relation, Eq. (b) can be arranged to give 

Or designating K _1+!2 

s- C P-2) 

Source: Joseph E. Shigley (1977). Mcclzmical Eqinechg Des$w, 3rd ed., McGraw-Hill Book Co., 

New York, NY. 

Figure 1. A sample page containing text, sketch, and equations. 

3 
then z = I$$ (8-3) 

10 12 14 16 

Spring index C 

FIGURE 8-2 

Values of the stress correction factors for round helical extension or compression 

springs. 

1.0 

0 2 4 6 8 

where K, is called a shear-stress multiplicationfactor. This factor can be obtained 

from Fig. 8-2 for the usual values of C. For most springs, C will range from about 

6 to 12. Equation (8-3) is quite general and applies for both static and dynamic 

loads. It gives the maximum shear stress in the wire, and this stress occurs at the 

inner fiber of the spring. 

Many writers present the stress equation as 

7 = KS (8-4) 

where K is called the Wuhl correctionfactor. * This factor includes the direct shear, 

together with another effect due to curvature. As shown in Fig. 8-3, curvature of 

the wire increases the stress on the inside of the spring but decreases it only 

slightly on the outside. The value of K may be obtained from the equation 

K _ 4C - 1 + 0.615 

-4c c 

(8-9 

or from Fig. 8-2. 

By defining K = K, K,, where K, is the effect of curvature alone, we have 

Kc=; V-6) 

s 

Source: Joseph E. Shigley 

New York, NY. 

Figure 2. A sample page containing text, graph, and equations. 

(1977). Mecharzical Engineering Design, 3rd ed., McGraw-Hill Book Co., 

4 
strengths for various wire sizes and materials.* But the availability of the scientific 

electronic calculator now makes such a tabulation unnecessary. The reason for 

this is that a log-log plot of the tensile strengths versus wire diameters is a straight 

line. The equation of this line can be written in terms of the ordinary logarithms of 

the strengths and wire diameters. This equation can then be solved to give 

SW=+ (8-10) 

where A is a constant related to a strength intercept, and m is the slope of the line 

on the log-log plot. Of course such an equation is only valid for a limited range of 

wire sizes. Table 8-2 gives values of m and the constant A for both English and SI 

units for the materials listed in Table 8-1. 

Although the torsional yield strength is needed to design springs, surprisingly, very little information on this property is available. Using an approximate relationship between yield strength and ultimate strength in tension, 

s, = 0.75&, (841) 

and then applying the distortion-energy theory gives 

s,, = 0.577s, (8-12) 

and provides us with a means of estimating the torsional yield strength S,,. But 

this method should not be used if experimental data are available; if used, a 

generous factor of safety should be employed, especially for extension springs, 

because of the uncertainty involved. 

Variations in the wire diameter and in the coil diameter of the spring have an 

effect on the stress as well as on the spring scale. Large tolerances will result in 

* See, for example, the second edition of this book: Joseph E. Shigley, ‘Mechanical Engineering 

Design,” 2d ed., p. 362, McGraw-Hill Book Company, New York. 1972. 

Table 8-t CONSTANTS FOR USE IN EQ. (S-10) TO ESTIMATE THE TENSILE 

STRENGTH OF SELECTED SPRING STEELS 

Material 

Size range, 

in 

Constant, A 

Size range, Exponent, 

mm In kpsi MPa 

Music wire” 

Oil-tempered wireb 

Hard-drawn wire’ 

Chrome vanadiumd 

Chrome silicon’ 

0.0040.250 0.10-6.5 0.146 1% 2170 

0.020-0.500 0.50-12 0.186 149 1880 

0.028-0.500 0.70-12 0.192 136 1750 

0.032-0.437 0.80-12 0.167 169 .2000 

0.063-0.375 1.6-10 0.112 202 2000 

’ Surface is smooth, free from defects, and with a bright lustrous finish. 

b Has a slight heat-treating scale which must be removed before plating. 

’ Surface is smooth and bright, with no visible marks. 

d Aircraft-quality tempered wire; can also be obtained annealed. 

’ Tempered to Rockwell C49 but may also be obtained untempered. 

Source: Joseph E. Shigley (1977). Meclzn~~ical hginecriq Dcsip, 3rd ed., McGraw-Hill Book Co., 

New York, NY. 

Figure 3. A sample page containing text, table, and equations. 
3. Usage of Tables 

Tables are ubiquitous in books and paper documents of all types. Although tables have been in use for centuries, research into tables and their 

properties is nonexistent. Children as young as six years old understand 

tables when they see a table of items at their favorite ice-cream shop. 

Little League baseball players understand the scores of their favorite 

teams when the scores are presented in tables (see fig. 4) (The Washington 

Post, 1996). In general, people use tables quite intuitively and very little 

instruction is ever provided on tables. Although the structure of tables 

can become very complex, as I will show in this report, mature adults 

learn the structure, properties, and relationships of items in a table by 

trial and error, sup&emented by intuition. _ 

Source: TIP Wuslzi~zgfo~r Post (1996). Monday, May 27, p C6. 

Figure 4. Status of baseball games on a given day shown by a number of tables. 

6 
Table 1. Example 

table showing two 

columns of data that 

hold a discrete 

relationship. 

Table 2. Example 

table showing two 

columns of data that 

hold a continuous 

relationship. 

An example of a simple table is a relationship between two columns of 

quantities, as shown in table 1. (The relationship can also be between two 

rows, but I use the column terminology for ease of discussion in this 

report.) 

The basic property of a table is that the item at the top of each column is a 

descriptor and not a value, while the rest of the items in a column are 

values. The two columns can also be considered two lists, usually with a 

correspondence relationship between items in each row. This table represents relationships in a discrete domain; that is, the relationships are 

meaningful only at the values given in the table, and extensions (interpolation and extrapolation) are neither possible nor meaningful. 

Table 2 gives another example of a simple table with two columns; once 

again, the first item in each column represents a descriptor, while the rest 

of the items are values, and there is a correspondence relationship between items in each column. Additionally, in such a table, one may be 

able to extend the relationship to other sets of values between those given 

in the table. Such a domain is continuous. For example, the value of Cost 

corresponding to an Area value of 2.5 can be calculated through interpolation. Several types of interpolations are available. (I do not discuss those 

various methods in this report.) Similarly, the value of Cost corresponding to an Area of 6.7 can be calculated by an extrapolation procedure. The 

important classification is that some tables represent continuous functional relationships and others represent discrete (point-wise) relationships. Database systems focus on the latter representation (a detailed 

discussion is presented in sect. 14). 

In popular vocabulary, tables are also called “tables of data.” People say 

“the data is stored in a table.” Such a statement implies that a table is 

nzerely a storage device. Ordinarily, a table (for example, either table 1 or 

table 2) stores data, and at the request of a user, the data are retrieved and 

presented. In such a system, storage is the primary purpose of tables. 

Database systems have been developed that essentially “store data” and 

return values upon a user’s request. Note the common expression, “data 

Office 

President 

Vice President 

Treasurer 

Secretary 

Person 

John Johnson 

Mary Gorton 

Catherine Weslock 

Timothy Wesbury 

Area cost 

1 3 

2 6 

3 11 

4 18 

5 27 

6 38 
is retrieved from a database.” In this sense (storage), databases and tables 

are used with synonymous meanings and for identical purposes. 

However, it is very important to distinguish between tables that represent 

data in continuous and discrete domains. Table 2 can be replaced by a 

functional relationship of the form Cost = 2 + (Area)2. 

Thus, an equation (a functional relation) and a table become mutually 

substitutable. This means that a table can be used where an equation can 

be used. In this sense, a table is more than a data storage device. This calls 

for a representation for tables beyond databases. 

Another example of tables that are much more than storage devices is the 

set of tables of logarithms. Before the advent of slide rules and calculators, tables of logarithms (see sample in table 3) were widely used for 

calculations in science and engineering fields. The logarithmic tables 

contained functional representations. Tables of values of reciprocals, sine, 

cosine, tangent, sinh, cosinh, tanh, and so forth are other examples of 

tables of functional relationships used directly to facilitate computations. 

As mathematics (specifically, algebra) progressed, people preferred to use 

equations rather than relying on tables of functions. However, this example shows that tables and equations can both represent the same 

functional relationship between variables in a given problem. Where 

equations are difficult to fit, tables of data may be the only recourse or 

else the best means, particularly when dealing with experimental data 

and empirical observations (such as baseball results). 

The history of steam tables shows the evolution of tables of data to equations for representing complex relationships. For two centuries, thermal 

engineers dealt with steam tables obtained through laborious experimental work. Steam has a thermodynamic state that is defined as a number of 

properties (pressure, temperature, specific volume, internal energy, 

enthalpy, and entropy). Given any two of these properties, the state is 

completely obtainable. Historically, steam tables (fig. 5) have been the 

main means of representing these relationships between thermal states of 

steam (Baumeister and Marks, 1964). With the advent of digital computing, the complex functional relationships between these properties have 

been captured in equations. Since these equations are extremely complex 

and do not reveal the relationships in the data, the steam tables and charts 

are very widely used. 

These arguments show that tables can be used in place of equations to 

represent functional relationships. Tables are more than storage devices. 

Tables are useful in doing calculations. For example, the ideal gas equation of state can be represented by a sample table, as shown in table 4. 
Table 3. Sample table of logarithms. 

l-18 MATHEMATICAL DATA AND CONVERSION TABLES 

LOGARITHMS 

Ifx=log,n, thenFX=n 

log fib = log a + log b 

log cdl> = log a - log b 

= -log b/o 

log l/n = -log n 

log n’ = c log n 

log vii = (I/c) log n 

log, r = 2.3626 log,0 x 

log,” x = 0.4343 log, z 

e = 2.718282 

LOGARITHMS TO BASE 10 

log l=O 

7T = 3.141593 

log,, r = .497150 

1.0 @xx) 004: 

1.1 0414 0453 

1.2 oi92 oaz 

1.3 1139 1173 

I.4 1461 1492 

15 li61 l79C 

1.6 2041 2w 

1.7 2304 233[ 

1.8 2553 257; 

1J 278X mc 

2.0 3010 3032 

2.1 3222 3243 

2.2 3424 3444 

2.3 3617 3636 

2.4 3802 382t 

2.5 3979 3991 

26 4150 4166 

2.7 4314 4330 

2.6 4472 4487 

2.9 4624 4639 

3.0 4771 4786 

2.1 4914 4928 

3.2 5051 5065 

3.3 5185 5196 

3.4 5315 5328 

35 5441 5453 

3.6 5563 5575 

3.1 5682 5694 

3.8 5798 5809 

33 5911 5922 

4.0 6021 6031 

4.1 6128 6138 

4.2 CL?32 6243 

43 6335 6345 

4.4 6435 6444 

45 6532 6542 

4.6 0628 6637 

4.7 6i21 6730 

4.8 6812 6821 

43 6902 6Y11 

5.9 6990 6998 

5.1 7076 7064 

5.2 7160 ;168 

53 2243 7251 

5.4 7324 7332 

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, 

, 

, 

I 

I 

I 

I 

I 

I 

45 4 

I7 21 25 

I5 19 23 

14 17 21 

13 16 19 

.2 15 ia 

!l 14 17 

I1 13 16 

IO 12 15 

9 I2 14 

9 I1 13 

8 11 13 

8 IO 12 

8 IO 12 

7 911 

7 911 

7 910 

7 810 

689 

689 

6 7 9 

679 

6 7 8 

5 7 8 

5 6 8 

5 6 8 

5 6 7 

5 6 7 

5 6 7 

5 6 7 

4 5 7 

4 5 6 

4 5 6 

4 5 6 

4 5 6 

4 5 6 

4 5 6 

4 5 6 

4 5 6 

445 

445 

3 4 5 

3 4 5 

3 4 5 

3 4 5 

3 4 5 

7 89 

29 33 37 

26 30 34 

24 28 31 

M 26 29 

tl 24 27 

20 22 25 

18 21 24 

17 20 22 

16 19 21 

16 18 20 

15 17 19 

14 16 18 

14 15 17 

13 15 17 

12 14 16 

12 14 15 

11 13 15 

I1 13 14 

I1 12 14 

LO 12 13 

IO 11 13 

10 11 12 

9 11 12 

9 10 12 

9 10 11 

9 10 11 

8 10 11 

8 910 

8 910 

8 910 

8 910 

7 8 9 

7 8 9 

7 8 9 

7 8 9 

7 8 9 

7 7 8 

6 7 8 

6 7 8 

6 7 8 

6 7 8 

6 7 8 

6 7 7 

6 6 7. 

6 6 7 

1 

-I 

Source: Hans Gartmann, ed. (1970). DC Lav~l El1~I’mwiu~ Hmdbook, 3rd ed., McGrawHill Book Co., New York, NY, l-18. 

9 
4-34 THERMAL PROPERTIES OF BODIES 9SD THERMODYNAMICS 

Table 27. Properties of Saturated Steam 

(From Keenan and Keyes, “Thermodynamic Properties of Steam”) 

(h, and 6, are measured from 32F) 

psi 

-- 

1.0 

1.2 

1.4 

1.6 

I.8 

2.0 

2.2 

E 

2.3 

3.0 141.43 

4.0 152.97 

162.24 

170.06 

176.35 

182.86 

188.23 

5.0 

6.0 

7.0 

8.0 

9.0 

10 193.21 

II 197.75 

I2 201.96 

13 205.83 

14 209.56 

14.696 212.00 

15 

16 

17 

13 

19 

213.03 0.016721 26.29 

216.32 0.01674! 24.75 

219.44 0.016771 23.39 

222.41 

225.24 

20 227.96 

21 230.57 

22 233.07 

23 235.49 

24 237.82 

25 

26 

27 

23 

29 

30 

31 

32 

33 

34 

35 

%“7 

38 

39 

101.74 

107.92 

113.26 

117.99 

122.23 

126.03 

129.62 

132.89 

135.94 

133.79 

240.07 

242.25 

244.36 

246.41 

248.40 

250.33 

252.22 

254.05 

255.84 

257.03 

259.23 

260.95 

262.57 

264.16 

265.72 

Liquid 1 Vapor Liquid 1 Evnp 

-_I_- ___I0.01614333.6 69.70 1036.3 

0.01616280.9 75.87 1032.7 

0.01618~243.0 81.20 1029.6 

0.01620214.3 35.91 1026.9 

0.01621 191.8 90.14 1024.5 

0.01623 173.73 

0.01624 158.35 

0.01626 146.38 

0.01627~135.73 

0.01629jl26.65 

93.99 1022.2 1116.2 0.1749 1.7451 1.9200 957.9 

97.52 1020.2 1117.7 0.1309 I.7311 1.9120 955.5 

100.79 1018.3 1119.1 0.1864 1.7183 1.9047 953.3 

103.33 1016.5 1120.3 0.1916 1.7065 1.3981 951.2 

106.63 1014.3 1121.5, 0.1963 1.6957 1.3920 949.2 

0.01630 113.71 109.37 1013.2 

0.01636 90.63 120.36 1006.4 

0.01640 73.52 

0.01645, 61.93 

0.01649 53.64 

0.01653 47.34 

0.01656/ 42.40 

0.016591 38.42 

0.01662; 35.14 

0.01665 32.40 

0.016671 30.06 

0.01670; 28.04 

130.13 1001.0 1131.1 0.2347 

137.96 996.2 1134.2 0.2472 

144.76 992.1 1136.9 0.2581 

150.79 988.5 1139.3 0.2674 

156.22 985.2 1141.4 0.2759 

161.17 

165.73 

169.96 

173.91 

177.61 

0.01672j 26.80 130.07 

/ 

0.016791 22.17 

0.01681) 21.03 

/ 

131.11 969.7 

184.42 967.6 

187.56 965.5 

190.56 963.6 

193.42 961.9 

0.016831 20.0891 196.16 

0.016351 

0.016871 

19.192' 193.79 

18.375 201.33 

0.0169li 0.01639] 

17.627 203.78 

16.938 206.14 

1 

0.01692i 

0.016961 0.01694' 

16.3031 208.42 

0.016981 

15.170715 210.62275 

14.663 214.33 

0.01699; 14.139i 216.86 

0.01707j 12.226; 226.18 

0.01708' ,l.89a1 

0.01709j 11.588' 

227.91 

229.60 

0.01711! 

0.01712' II.2941 11.015, 

231.26 

232.39 

0.017l41 lO.75Oj 234.48 

932.1 1143.3 0.2335 1.5041 1.7376 911.1 

979.3 1145.0 0.2903 1.4897 1.7300 907.3 

976.6 1146.6 0.2967 1.4763 1.7730 904.8 

974.2 1148.1 0.3027 1.4633 1.7665 901.9 

971.9 1149.5 0.3083 1.4522 1.7605 399.3 

970.3 1150.4 0.3120 

960.1 1156.3 0.3356 

958.4 1157.2, 0.3395 

956.3 1153.q 0.3431 

955.2 1159.0' 0.3466 

953.7 1159.8 0.3500 

952.1 1160.6 0.3533 

950.7 1161.3 0.3564 

949.3 

947.9 

1162.0, 0.3594 

1162.71 0.3623 

946.5 1163.41 0.3652 

945.3 

944.0 

942.3 

941.6 

940.3 

939.2 

938.0 

936.9 

935.3 

934.7 

1150.8' 0.3135 

1152.0 0.3184 

1153.1 

1154.2' 

0.3231 

0.3275 

1155.3 0.3317 

1164.1 0.3680 

1165.41 1164.71 

0.3707 

0.3733 

1166.0' 0.3758 

1166.5 0.3733 

1167.1 0.3307 

1167.6 0.3331 

1163.21 0.3854 

1168.7] 0.3376 

1169.2; 0.3393 

974.6 

970.3 

966.7 

963.5 

960.6 

1.5333 1.8057 913.4 

1.5203 1.7962 914.6 

I.44461 

1.4213i 

1.4123 

I.40431 

1.3962l 

I.3885 

i.33ili 

1.3740' 

1.3672' 

1.7566 

1.7549 396.7 

1.7497 394.3 

1.7449 392.0 

1.74031 a89.9 

1.7360 887.3 

1.7206 

1.7172 

397.5 

835.3 

383.9 

332.0 

380.2 

373.5 

I.36061 I.71391 376.3 

I.35441 1.7108, 875.2 

1.3484 1.7078 373.6 

1.3425! 1.7043 872.1 

1.3368, 1.7020 870.5 

, I.33131 1.6993' 

1.3260 I.69671 

I.32091 1.6941 

1.3159! I.69171 

1.3110/ 1.6893; 

I 

l.3063! I.65701 862.3 

I.30171 1.6848: 861.0 

1.2972 1.6826 

I.29291 1.6805/ 

839.3 

358.5 

1.2336' 1.6734/ 857.2 

869.1 

867.7 

866.3 

864.9 

363.5 

Source: T. Baumeister, and L. S. Marks, eds. (1964). Mechanical Engineerrs' Halzdbook, 6th 

ed., McGraw-Hill Book Co., New York, NY. 

Figure 5. A sample table showing properties of saturated steam. 

10 
Table 4. Tabular 

representation of 

temperature-volumepressure relationship 

for an ideal gas. 

Table 1. Specific volume of air (ft3). 

11 
4. Ends and 

12 

Means 

The fact that tables can be used to represent functional relationships has 

important implications for software developers. Currently, most software 

methods generate tables that can be used as records and not as dynamic 

relationships, Contrast this with tables in books-a scientist can readily 

use tables from books (in paper) either to read or to be used as a relation 

in his or her computing. Regrettably, since computer tools are not available that treat tables as functional relationships, their reuse is limited and 

circuitous. 

Coad and Yourdon (1991) caution software developers that if the application of a software engineering method produces a monument of paper, 

then something is wrong-in the method, in the application of the 

method, or perhaps both. They lament, “if we lose sight of people and 

begin producing charts, diagrams, and piles of paper as ends [italics 

added] unto themselves, we fail to effectively communicate.” 

By developing structures and methods to represent tables as nzedns, as 

well as elzcls, we will be able to facilitate seamless reuse of tables in computer systems. In such a case, tables will become nzedns for communication among people, understanding by people, and further generation of 

knowledge by people. That is, tables can be “copied and pasted” in 

developing new computer programs. 
5. Treatment of Tables in Text Formatting Tools 

Early word processing software tools could not handle tables. Subsequent 

word processors recognized the need for tables to be embedded in text 

documents. Initially, tables were provided as simple rectangular arrays. 

Recent developments cater for some basic variations in tables in text 

processing tools both for electronic (World Wide Web) and paper (word 

processing) documents. However, all these are display devices only. 

Morris (1996, p 79) describes how tables are formatted and added as a 

new feature of HTML (HyperText Markup Language, the primary language of the World Wide Web) to make HTML a true publishing medium. 

As is common with HTML documents, tags are used to define a table and 

its components. A table is divided into rows and cells. Techniques are 

defined to format text in table cells. However, these formatting rules are 

merely to represent tables for display and visualization. Creation and 

editing of tables are permitted, but no other data manipulation is possible 

at present. That is, no calculations are made with tables. Lemay (1996) 

also discusses formatting HTML tables for use on the World Wide Web. 

The emphasis is on creating tables at the transmitter’s terminal (or server) 

and displaying them at the receivers’ (client) terminals. The Web browser 

Netscape features table heading cells and data cells. Lemay also suggests 

using lists, images, and preformatted text as alternatives to tables. 

Microsoft Word 95 accommodated in its “Autoformat” 38 different table 

formats. It also allows for columns to be split (see table 5). However, the 

purpose of these operations is again to display a table in a text document 

in a natural fashion from a human interface point of view. But no methods exist to support the use of tables as functional relationships. 

In the last few years, however, tables in word processors have become 

capable of performing simple calculations. For example, the totals of 

several items in a column (or a row) can be computed automatically as 

the items in the table are entered. This is somewhat similar to the capabilities in a spreadsheet. However, these features in tables are minor compared with the capabilities of tables in Natural Computing as described in 

this report. 

Natural Computing also provides for use of spreadsheets with programming interfaces. Additionally, in Natural Computing (Karamchetty, 2000), 

spreadsheet strips are independent objects that behave like spreadsheets 

while being small in size, with positive implications on memory requirements and performance. 

In this report, my focus is on the use of tables from the perspective of 

calculations. Therefore, the emphasis is on the representation of a table for 

calculation rather than for its presentation as part of a document. 

Table 5. A table in a 

word processor Monday Tuesday Wednesday Thursday 

showing an element AM PM 

spanning two - - - - 

columns. - - - - - 

- - - - - 

13 
6. Informal 

Figure 6. Example of a 

simple table with its 

parts identified. 

14 

Survey and Analysis of Practical Tables 

In this section, I provide a number of examples of tables from various 

textbooks, handbooks, brochures, and newspapers to demonstrate the 

richness and complexity of data, information, and functional relationship 

representations in practical tables. These examples allow me to identify 

and describe characteristics and properties of tables. 

A table represents characteristics and values in a domain. A twodimensional rectangular array is traditionally used to represent a table. 

But since a table contains a set of characteristics and their values, the 

representation should recognize this distinction by separating the table 

header, which contains the characteristics, from the data body, which 

contains the values. A table can be identified by a table caption, which 

gives it a unique identity in the document. In documents such as books, 

we find lists of tables, which bring together all the table captions to one 

location (generally with a table of contents (TOC) giving the page numbers of table locations). Thus, a simple table will consist of a table caption, 

a table header, and a table data body. The table caption is a string. The 

table header is a row (list) containing strings. The table data body is a 

two-dimensional matrix representing values of a given type (a number, a 

string, or a graphic symbol). Figure 6 is an example of a simple table with 

its parts identified. 

Compare this simple table with the sample table in figure 7 (Mechanical 

Engirzeeus’ Handbook, Baumeister and Marks, 1964). In the sample, three 

types of values are used in the table data body: a string, a real number, 

and a range of real numbers. Such a combination of data types does not 

allow the use of an array in computer representation (since arrays represent data of the same type). Of course, one possible representation is an 

array of pointers, with each pointer pointing to a given data type (string, 

real number, and range). 

Caption - Table 7-l. Thermal properties.* 

Header 

(psia) (deg F) 

Body 

100 212 5.4 

k 

150 239 4.3 

I 200 320 3.6 I I 

250 360 3.1 

I 300 410 

Footnote - h *Applicable to real gases. 

2.7 
Figure 7 shows that table header elements are usually complex data 

forms. In this sample, the second column label includes the descriptive 

name (absolute roughness) of the item, its symbol (E), and its units (feet). 

From a textual visibility standpoint, representation as a string will suffice. 

However, a different and more elegant representation is needed to allow 

these three parts to be identified and accessed uniquely. 

Figure 8 shows a sample table with data types that are numbers (both 

integers and a floating point), ranges of numbers (with range indicated by 

” ” as well as “to”), strings, and an algebraic expression (an equation). - 

This table also has another feature: several footnotes (Baumeister and 

Marks, 1964). 

Figure 7. A sample Table 4. Absolute Roughness Classification of Pipe Surfaces for Selection of Friction 

table with a variety of Factor f in Fig. 18 

data types. Commercial Pipe Surfaces (Nrw) Absolute Roughness c. Ft 

Glass, draw” brass. copper, lead.. _. Smooth 

Wrought iron. ntcel.. _. _, _. __. __ 0.00015 

Asphalted cast iron.. 0.0004 

Gslvanizediron............................................... 0.0005 

castiron..................................................... 0.00085 

Woodstsve........................................,.,....... 0.0006-0.003 

Conrrete..................................................... 0.001-0.01 

Rivctedstr~el................................................. 0.003-0.03 

Source: T. Baumeister and L. S. Marks, eds. (1964). Meclmicd EIZ$IKWS 

Hadbook, 6th ed., McGraw-Hill Book Co., New York, NY. 

Figure 8. A sample 

table showing 

distinct data types. 

3-78 MECHA?;ICS OF LIQUIDS 

Table 6. Loss of Head in Flow of Water through Elbows, Meters, Etc. 

(Also expressed in equiralcnt lengths of straight pip% to cause equal loss) 

Loss of head a% Equivalent length 

Nature of resistance 

a derimnl or of Rtrainht pip% 

multiple of V/20 expresacd in 

i (nnfe avg VaIllea) pipe dinmctcrs 

Square-edged entry. Upstream md of pipe flush with inside 

face of reservoir wall.. . . . 0.50 20 

Entry like Borda’a mouthpiece (see Orifices) 1.00 40 

Rounded entry or very large radius bends. o-o.05 Z%Kl 

* 90 deg curve%. smooth. MIIIL ionidv dim” as pipe: 

Center-line radius = dinm of I~ipe.. . 0.50 20 

Center-line radius = 2 to 3 ditllu _. 0.25 10 

90 dep elbows. oomr”o” %%nw end. short turn (experimenti on 

y8 to(;in.clls)........................................ 0.75 30 

Tea. common %cr%w end, full size brunch (experiments on 1 to 

4i”.b%S) I .50 60 

Square elbow (intersection of tw_lrrylindvr%1.. 1.25 50 

Obtuse-analed elbows. deflection in I),,= - a deg (leas than YO 

dee,.................................................. I. 25 X (ni90)z 50 x (o/90)2 

Water 1ueterst 

Disk or wobble type., .I 

Rotary (d&k of %fnr or cop-vi& %h%pc R% piston). 

Reciprocating piston (like % piston pump). 

Turbine wheel typ+ (double. flow. bnlmrrd). _: 

3.4 to 10 135-400 

10 400 

600 

5-? 5 200-300 

For pipe orifices in diaphragms see p. It?-1Y. 

*For complete? mxnmary and details, %%e Pigott. Trans. ASME. 1950 and IYGG. 

t Different makes of meters and diflctrent %izcs of the %%m% make wry considerably. 

Source: T. Baumeister and L. S. Marks, eds. (1964). Mcchical Eugimzers’ 

Hmdbook, 6th ed., McGraw-Hill Book Co., New York, NY. 

15 
Figure 10. Example of 

complex case and 

switch statements 

required to replace a 

table such as that in 

figure 9. 

16 

The sample table in figure 9 reveals another highly interesting use of 

tables in practical applications. Data elements in this table contain algebraic expressions that are the equivalent of several case and switch statements, as shown by pseudo-code in figure 10. It is excitingly obvious that 

such tables represeni considerable information very elegantly. 

Table 7. Values of the Hydraulic Radius m for Various Cross Sections 

(m = cross-sectional area i wetted perimeter) 

Form of oross section I Value of nz 

Circle, flowing full, also half full. . 

Square, no top (depth = width = d). . 

Half square (width = 2d, depth = d). 

Trapezoidal channels (bottom width = b; depth = d): 

Half regular hexagon. side slopes 60 deg. 

Channel with 45 deg side slopes.. 

Channel with side slopes 1% hor to 1 vert.. 

Channel with side slopes 2 hor to 1 vcrt., 

Wide. shallow stream 1.. 

............. 

............. 

............. 

............. 

............. 

d/4 

d/3 

d/2 

d/2 

(bd + d9l(b + 2&d) 

(bd + I.M2)/(b + 3.61d) 

(bd f Zd?/(b + 4.47d) 

d bpprox) 

1 Hence the term “hydraulic mean depth.” sometimes used for m. 

Source: T. Baumeister and L. S. Marks, eds. (1964). Mechanical Engineers’ 

Handbook, 6th ed., McGraw-Hill Book Co., New York, NY. 

Switch (form of cross section) { 

Case (circle, flowing full, also half full): 

, Characteristic from 

table header 

{m=d/4; t 

break: 

1 

Case (square, no top (depth = width = d): 

( ,,I = d/3; 

break; 

and so on 

I 

- Value from 

table body 

Relationship equation 

given as value 

in table body 
Figure 11. A sample 

table showing a 

conditional range 

value in an element. 

In the sample table in figure 11, the fourth column represents a temperature range. However, this range is not an output or value range as in the 

previous sample (see fig. 7). It is a conditional mzge. This is a clever way of 

representing a number of “if statements ” in a column. The equation given 

in column 3 is applicable only if the “Temp range, deg R,” is within the 

range given at any element. This interesting feature leads to the representation of a table element as a data output value, or as a conditional range 

for checking. In other words, the table takes multiple data values. For 

example, if the gas is oxygen and if the temperature range is between 540 

and 5000, then the equation is as given in column 3. This is indeed a very 

intriguing way of representing calculations through tables. 

Table 17. Specific Heats of Gases at 1 Atm 

GIL9 Equation for Cp in Btu per mol source 

I-I --- i-lOxygen. 

Nitrogen.. 

540-5000 

0 05(T - 4000) 

+ ~-Too0 _) 1 5000-9000 / (I 

I I 540-5000 a 

Carbon monoxide. 1 CO 1 9.46 - (“F) + (““g) 1 540-5000 1 0 

Hydrogen.. HI 

Water. HIO 

Carbon dimide. CO% 

Methane.. CII4 

5.76 + (“$$) + (5) 

5.76 + (“e) + (“) 

\/T 

_ ( o.33v-04000)) 

19.66 - (z) + (y) 

16.2 _ (6v) + (w) 

4.22 + 8.21 1 X IO-‘T 

27.0 - ‘4~!!!?. 

T 

Ethylene.. CIH, 6.0 + R.33 X IO-‘T 

Ethnne. Cl116 6.6 + 13.33 x IO-‘T 

Ethyl alcold.. CrttsO 4.5 + 21.1 x IO-JT 

m?ttayt 31cot101. CHAD 2.0 + 16.67 X IO-IT 

Benmw. Cdl.. 6.5 + 28.9 X IO-ST 

Octnne. GHn 14.4 + 53.3 x IO.‘T 

Dodecnne C,ZHIG , 19.6 + 80.0 X IO-IT 

540-4000 

4000-9000 

540-5000 

540-6300 

492-1800 

1800-5940 

n Sweigert and Beard&y, Empirical Specific Heat Equations Based upon Spectroscopic Data, Da. School 

Tech.. State Eng. Ezpr. S&x. Bull. 2. IORE. 

b Schwnrz. Die Spez&cbm Wiirmen dw Gase als HiIfawertc zur Bereclmung van Gleicbgenicbtcn. Arch. 

Eisenhiiffcnw.. 9, 1936. p. 389. 

0 Parks and Huffman. ACS. Mon. 60. 1932. 

* Approximate. An equation bssed ou the most recent data is given by Keyes in J. Chem. Plrys.. 16, 

Aug. 19J7. p. 602. 

720-1400 

720-1440 

680-l 120 

680-l 100 

520-l 120 

720-1440 

720-1440 

c 

c 

Source: T. Baumeister and L. S. Marks, eds. (1964). Meclmical En~inccrs’ 

Hmdbook, 6th ed., McGraw-Hill Book Co., New York, NY. 

17 
Figure 12. Example 

table where elements 

develop branches as 

they progress along 

columns. 

18 

In my presentation of table samples so far, the table header and the table 

data body are rectangular; i.e., a number of rows and columns form a 

regular rectangular array Now consider the table in figure 12. Note that 

both columns 7 and 8 in the table header develop branches. First, the 

element “Specific heat per lb . . . ” branches into cP and c,. Next, the element “Heat capacity per cu ft. . .” develops the same branches, cP and c,. 

The table header begins with 9 columns and, with these branches, ends 

with 11 columns (the same number as the table data body). At the junction of the table header and body, the number of columns in both the 

table header and body is the same. The table in figure 12 is an example of 

columns branching into two or more columns. 

The most significant effect of the branched columns is that a simple 

rectangular array representation for tables is not adequate. We must be 

able to produce table header and table data body representations that will 

allow for any number of columns developing branches. To accommodate 

such irregular shapes (nonrectangular) for a table header and a table data 

body, we need to adopt a different data structure. I introduce the concept 

of a cell cage to accommodate this requirement (see sect. 7). 

As a side point, figure 12 also has blanks in the table data body represented by several dots (“. . . “). Yet another feature of tables-footnotescan also be observed in this example. The first column of the table data 

body has an entry (Steam) (about midway down) with a symbol c referring to a table footnote. Footnote symbols require another addition to the 

Table 23. Properties of Gases 

Helium.. . . 

Argon. . 

Air.. 

. . . A 

Oxygen.. 

Steamc.. 

Carbon dioxide.. . . . 

Sitrous oxide., 

Sulpbur dioxide. 

Ammonia. . 

_4cetylene, 

Methyl chloride.. 

Xethane. . . . 

Ethylene.. . 

Source: T. Baumeister and L. S. Marks, eds. (1964). Mechanical Engiwers’ 

Handbook, 6th ed., McGraw-Hill Book Co., New York, NY. 
Figure 13. A number 

of columns develop 

variable numbers of 

branches or 

multiples. 

item/value representation. The cell must be a composite, having both the 

value “Steam” and a footnote symbol. The table will then carry a footnote 

linked to that symbol. Figure 13 also shows a sample table where a number of columns develop multiple branches. 

P34 

Ahs 

press, 

psi 

1.0 

I.2 

1.4 

1.6 

1.8 

2.0 

2.2 

2.4 

2.6 

2.8 

3.0 

4.0 

5.0 

6.0 

7.0 

8.0 

9.0 

IO 

II 

I2 

13 

I4 

14.696 

15 

16 

I7 

I8 

19 

20 

21 

:: 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

THERMAL PROPERTIES OF BODIES dlr;D THERMODTNAMICS 

Table 27. Properties of Saturated Steam 

(From Kcenan and Keyes, "Tllermodynrunic Properties of Steam") 

(h,snd s/we meowred from 32F) 

Enthslpy I Entropy 

Internnl 

mcray 

dcg P i I I 

1 Liquid 1 Vapor Liquid 1 Evap Vapor Liquid 1 Evap ’ Vapor Eva11 

___~_-;___1----._---- __~-- 

69.70 1036.3 1106.0~ 

1110.8 I 

0.1326 974.6 

75.87 1032.7 1108.6 0.1435 

I.84561 1.9782' 

I.81931 I.96281 970.3 

81.20 1029.6 0.1528 966.7 

85.91 1026.9 1112.8 0.1610 1.7776 1.9386 963.5 

90.14 1024.5 1114.6 0.1683 

I.7971 Ii 1.9493i 

1.7605 1.9288 960.6 

126.08 0.01623/173.73 93.99 11022.2 1116.2 0.1749 1.7451 1.9200 957.9 

129.62 

1 

0.01624:158.85 97.52 1020.2 1117.7 0.1809 1.7311 1.9120 955.5 

132.89 0.01626'146.38 100.79 1018.3 1119.1 0.1864 I.7183 1.9047 953.3 

103.83 1016.5 1120.3 0.1916 1.7065 1.8981 951.2 

106.68 1014.8 1121.5 0.1963 1.6957 1.8920 949.2 

120.860937 1006.4132 1122.673 0.200819 I.64271 1.6855' 1.8625363 ;:::: 

162.24 

170.06 

176.85 

182.86 

183.28 

193.21 

197.75 

201.96 

205.88 

209.56 

212.00 

213.03 

216.32 

219.44 

222.41 

225.24 

227.96 

230.57 

233.07 

235.49 

237.82 

240.07 

242.25 

244.36 

246.41 

243.40 

0.01640! 

0.016451 

73.52 

61.98 

0.01649 53.64 

0.01653 47.34 

0.01656 42.40 

130.13 

137.96 

144.76 

150.79 

156.22 

161.17 

165.73 

169.96 

173.91 

177.61 

0.01659 38.42 

0.01662 35.14 

0.01665 32.40 

0.01667, 30.06 

0.016701 28.04 

I 

0.01672i 26.80 

O.Ol672i 26.29 ' 181.11 

0.016741 24.75 134.42 

0.01677 23.39 137.56 

0.01679 0.016311 22.17108 ;;;::; 

180.07 

0.016831 

0.01687' 0.01685l 

20.089' 

18.3759192/ 

196.16 

201.33 198.79 

0.016891 17.627 203.78 

0.01691 16.938 206.14 

0.01692 16.303 208.42 

0.01694 15.715' 210.62 

0.01696 15.170' 

0.01698' 14.663; 

212.75 

214.83 

0.01699 14.189; 216.86 

: 259.28 / 0.01703 11.898' 227.91 

i 260.95 

I 262.57 

/ 0.01709' Il.588 229.60 

/ 0.01711' Il.294 231.26 

I 264.16572 / 

232.89 

i 0.01712; 0.01714! 11.015! 10.75Oj 234.48 

1001.0 

996.2 

992.1 

988.5 

985.2 

~ 982.1 

979.3 

976.6 

974.2 

971.9 

970.3 

969.7 

967.6 

965.5 

963.6 

961.9 

960.1 

958.4 

956.8 

955.2 

953.7 

952.1 

950.7 

949.3 

947.9 

946.5 

945.3 

944.0 

942.8 

941.6 

940.3 

939.2 

938.0 

936.9 

935.8 

934.7 

1131.1 0.2347 

1134.2 0.2472 

1136.9 0.2531 

1139.3 0.2674 

1141.4 0.2759 

1143.3 0.2835 

1145.0 0.2903 

1146.6 0.2967 

1148.1 0.3027 

1149.5 0.3083 

1150.4 0.3120 

1150.8' 0.3135 

1152.0 0.3184 

1153.1 0.3231 

1154.2 0.3275 

1155.3 0.3317 

1156.3 0.3356 

1157.2 0.3395 

1158.1 0.3431 

1159.0 0.3466 

1159.8 0.3500 

1160.6 0.3533 

1161.3 0.3564 

1162.0 0.3594 

1162.7; 0.3623 

1163.4i 0.3652 

1.5820 1.8292 

1.5586 

1.5383 

I.81671 

1.8057, 

1.5203 1.7962' 

I.5041 1.7876 

1.4897 1.7800' 

1.4763 1.7730 

1.4638 1.7665 

1.4522 1.7605 

1.4446 1.7566 

1.4415 1.7549 

1.4313 1.7497 

1.4218 1.7449 

I.4128 1.7403' 

1.4043 1.7360' 

1.3544 1.7103/ 

1.3484, I.70781 

1.3425 I.70481 

l.3363/ 1.7020/ 

933.0 

927.5 

922.7 

918.4 

914.6 

911.1 

907.8 

904.8 

901.9 

899.3 

897.5 

896.7 

894.3 

892.0 

889.9 

887.8 

835.3 

883.9 

882.0 

330.2 

878.5 

876.8 

075.2 

073.6 

872.1 

870.5 

869.1 

867.7 

866.3 

864.9 

863.5 

862.3 

861.0 

859.8 

853.5 

857.2 

Source: T. Baurneister and L. S. Marks, eds. (1964). Mcclmicnl Enginccss’ 

Hmdbook, 6th ed., McGraw-Hill Book Co., New York, NY. 

19 
The sample table in figure 14, which shows properties of superheated 

steam, is an example of a table with both a row header and a column 

header. In the column header, one column (Temperature of steam, deg F) 

develops nine branches. Each row in the row header (20,40,60, and so 

on) develops three branches. A number of elements are blank (indicated 

by a series of dots), indicating no meaningful values, or indicating that 

table values have bounds. 

Figure 14. A sample 

table with both a 

column header and a 

row header. 

Table 28. Superheated Steam Tables 

(Abstracted from Keenan and Keyes, "Thermodynamic Properties of Steam") 

(n = specific volume,ou ft perlb; h = enthalpy. Btu perlb; s = entropy) 

Temperature of stesm,deg F 

(Saturation 

tem~,degF) 

(22?96) 

340 380 1 420 1 460 500 I 550 600 700 

23.60 24.82 26.04 27.25 28.46 29.97 31.47 32.97 34.47 

1210.8 1229.7 1248.7 1267.6 1286.6 1310.5 1334.4 1358.6 1382.9 

1.8053 1.8285 1.8505 1.8716 1.8918 1.9160 1.9392 1.9671 1.9829 

- 

II 

s 

11.684 12.315 12.938 13.555 14.168 14.930 15.688 16.444 17.198 

1207.0 1226.7 1246.2 1265.5 1284.8 1309.0 1333.1 1357.4 1381.9 

1.7252 1.7493 1.7719 1.7934 1.8140 1.8385 I.8619 1.8843 1.9058 

7.708 8.143 8.569 8.988 9.403 9.917 

1203.0 1223.6 1243.6 1263.4 1283.0 1307.4 

1.6766 1.7135 I.7250 1.7470 1.7678 1.7927 

5.718 6.055 6.383 6.704 7.020 7.410 

1198.8 1220.3 1240.9 1261.1 1281.1 1305.8 

1.6407 1.6669 1.6909 1.7134 1.7346 1.7598 

10.427 10.935 

1331.8 1356.3 

1.8162 I. 8388 

7.797 

1330.5 

II.441 

1380.9 

1.8605 

4.521 

1194.3 

1.6117 

1.7836 

8.180 8.562 

1355.1 1379.9 

1.8063 1.8281 

4.801 5.071 5.333 5.589 5.906 6.218 6.527 

1216.8 1238.1 1258.8 1279.1 1304.2 1329.1 1354.0 

1.6391 1.6639 1.6869 1.7085 1.7340 1.7581 I.7810 

6.835 

1378.9 

1.8029 

3.964 4.195 4.418 4.636 4.902 5.165 5.426 5.683 

1213.2 1235.3 1256.5 1277.2 1302.6 1327.7 1352.8 1377.8 

1.6156 1.6413 1.6649 1.6869 1.7127 1.7370 1.7601 1.7822 

(26?25) 

v 

h 

s 

(29?7l) 

(31821)03) 

100 

(327.81) 

120 

(341.25) h 

s 

(353.02) 140 

160 

(363.53) 

I80 

(373.06) 

2.914 3.098 3.273 3.443 3.640 3.849 4.048 4.244 

1205.5 1229.3 1251.6 1273.1 1299.3 1325.0 1350.4 1375.7 

1.5766 1.6042 1.6291 1.6519 1.6785 1.7033 1.7268 1.7491 

2.563 2.732 2.891 3.044 3.230 3.411 3.586 3.764 

1201.4 1226.1 1249.1 1271.0 1297.6 1323.5 1349.2 1374.7 

1.5596 1.5884 1.6139 1.6373 1.6642 1.6894 1.7130 1.7355 

220 

(389.86) 

260 1 u ._..., i ____.. 

(404.42) h .._.._ 1 ___.._ 

/ I .3 ! 

I.5513 1.6638 1.7675 1.8891 2.005 2.118 2.227 

1204.8 1232.5 1257.6 1286.8 1314.7 1341.8 1368.3 

1.5126 1.5434 1.5701 1.5998 1.6268 1.6517 1.6751 

350 

(431.72) 

1.3984 1.4923 1.6010 1.7036 1.8021 1.8980 

1224.8 1251.5 1282.1 1310.9 1338.5 1365.5 

1.5197 1.5481 1.5792 1.6070 1.6325 1.6563 

...... 

...... 

I.1978 1.2851 1.3843 1.4770 1.5654 1.6508 

1216.5 1245.1 1277.2 1306.9 1335.2 1362.7 

1.4977 1.5281 1.5607 1.5894 1.6155 1.6398 

...... 

...... 

...... 

...... 

/ / 

- 

Source: T. Baumeister and L. S. Marks, eds. (1964). Mechanical Engineers’ 

Handbook, 6th ed., McGraw-Hill Book Co., New York, NY. 

20 
Figure 15. A sample 

table showing ranges 

with value 

correspondence. 

This table has another extremely interesting feature. Note the characteristic value in the column header, “Pressure, psia (Saturation temp, deg F).” 

In the corresponding row values, note two numbers, one without and the 

other with parentheses. I refer to this set as a value and a parenthesis (or 

bracket) value. The pressure values are represented by the values in the 

open, and the saturation temperature values are placed in parentheses. 

From a computer representation point of view, this feature should be 

considered when representations for cells are developed (see sect. 8). 

The sample table in figure 15 presents interesting examples of the use of 

ranges. As the footnote to the table indicates, when a range is given in an 

element, the first and the second numbers correspond, and linear interpolation is permitted. 

‘I’able 1. Emissivity of Surfaces 

Highly polished.. . 

Polished. . . . . 

Rough plnte. _. . . 

Oxidized z.t 1110 F. 

Oxide.. . 

Alloy 7.sT.. . 

7557, revated heating . . 

Bnu: 

Hi&y polhhed 

Rolled plate. natural. 

Rolled. cousc 

emeried. 

OxidiredatlllOF.... 

Chromium. . . . . . 

copper: 

Electrolytic. polished. 

Comm’l plnte. 

pa&he&. . 

HesudatlllOF..... 

Thick oxide costing.. 

Cuprous oxide.. . hidten capper.. 

Dow metal. clesned. 

bested . 

Gold. highly polished.. Iron and steel: 

Pure Fe, polhhed 

Kroupht iron. 

polished. . 

Smooth sheet iron. . 

Rusted p!nte. 

Smooth oxidized iron. 

Strongly oxidized bidten iran nnd steel. 

hd: 

99.86%. unoxidized.. Gray oxidized. 

Oxidized br390 F.. _. Jierc”~. pure clem 

MAybdenum filnmenr. 

hfooel metal. K;;w. 

Wuhed. nbrusve soap 

Repeared belting 

Kickel nnd alloy. 

Electmlytic. polished 

Electroplnted. wt 

poliiberl.. wire.. 

Plate. axid. t.t 1110 F. 

Sickel oxide.. _. _. 

Capper-nickc!. 

polished 

~iekel~silrcr. polished 

Sickclin. gray oxide. 

440-1070 

73 

78 

390411a 

530-1520 

75 

450-900 

497-710 

11 

72 0.20 

390-Ill0 0.61-0.59 

loo-1000 0.08-0.26 

I76 0.02 

39Zila 

77 

1470-2010 

1970-2330 

4s(c750 

440-l 160 

350-1800 0.05-0.37 

100-4110 0.28 

1650-1900 a.55-0.60 

6i 0.69 

160-P@O 0.78-0.82 

100-4.90 0.95 

2730-1220 0.40-0.45 

68 

368-1844 

390-1110 

1200-2290 

212 

212 

70 

0.11-0.19 

0.63-0:26 

0.10 

\0.22-0.16 

0.03-0.04 

0.06 

0.030 

0.57-0.57 

0.78 

0.66-0.54 

0.16-0.13 

0.24-0.20 

0.02-0.40 

O.Ob-o.oa 

0.28 

0.63 

0.09-0.12 

0. IO-O.29 

0.17 

0.46-0.65 

I 0.06 

I 0.14 0.26 

Surface 

Nichrome wire. bright 

Nicbrame wire. o?iiJ.. 

ACI-HW((BOSi.l?Cr), 

&m black OX. CIIBI. 

Plntinum. polis!led plnte 

Silver. pure polisbcd. 

StainIns steeln: 

Type 316. cleaned. 

316, repeated hesting. 

304, .I? brat QSO F.. 

310. foroacescrrice.. 

Allegheny C-r. polished 

Tanralum filsment.. 

Thorium oxide. . 

Tin. bright.. _. . 

Tungsten. aged fiiament. 

Zinc. 9Q.l?$. eonm’l. 

polished.. 

Galv.. iron. briqht... 

Galv.. gray oxid.. . 

120-1830 0.61-0.79 

120-930 0.95-0.98 

520-1045 0.89-0.82 

440-2960 o.os-0.17 

440-1160 0.02-0.03 

75 0.28 

450-1600 0.57-0.66 

420-980 0.62-0.73 

47.0-980 0.90-0.97 

212 0.13 

24255430 0.194-0.33 

530-1220 0.58-0.21 

76 0.04-0.06 

SO-6000 0.03-0.35 

440-620 0.05 

02 0.23 

75 0.28 

Refractories. Building ?&terials. l’tiin~. MAC. 

Alumina. 60~ grain rire.. 

Alumina-siiics cnn*‘g.. 

0.490 Fe,O,.‘. :. 

1.7% FaO,.. . 

2.9Yo FezG I..... . Al painta (vary xirh 

.m’t lacquer body. ( 

age), . . ., 

.tibestor.............. 

Candle soot: ImnvbIilzek- , 

nwer gl3se 

Carbon plate. heated.. , 0x1 Inyers: 

Lubeoil. 0.01”~ pbl. 

Ni _..,.._....,.... j 

Lineed. l-2 colts on t 

Al. 

Rubher. soft gwt recl.imed.. _. 

Misc. I: rbiny black 

1 

lacquer. planed aik. 

white enemel. aerpentine. gypsum. wiite 

enamel paint. roo6ng 

paper. lime pItiter. / 

blsck matteshellac. ( Mix. II: glued porceIsin. wbire paper. 

fused quartz, p&bed 

mnrble. rousb red 

brick, smooth ~lata. 

hxd glossy rub’bcr, 

flat blnek lacquer. XICr. e1eclroqrapllitc 

18552850 0.39-0.28 

1850-2850 

,....,..... 0.61-0.43 

,.__.,.._.. 0.73-0.62 

..,....._.. 0.78-0.68 

212 0.27-0.67 

100-700 0.93-0.9s 

2;n-:o”,o ,oi9~,?aoi~l 

I -’ 

68 0.82 

68 0.160.57 

76 0.86 

70 ! 0.87-0.91 

I 

! 

I 

70 j 0.92-0.96 

/ 

Source: T. Baumeister and L. S. Marks, eds. (1964). Mrchaizical En#wcrs’ 

Hamfbook, 6th ed., McGraw-Hill Book Co., New York, NY. 

21 
Figure 16 shows a sample table for which dimensions or attributes are 

described by means of a sketch. This type of table is useful in generating 

the drawings of a part or machine component, as the table provides the 

flange sizes for a given load and nominal pipe size. A number of additional features are observable from the structure of this table. Notice the 

notes in the body of the table that redirect the calculations to a different 

part of the table. 

Figure 17 shows a sample table from a catalog of a plastics manufacturer. 

The size of the container is shown as L x W x D. But three separate columns are not used. In terms of cell representation, the column header 

should be represented as a composite cell containing the characteristic 

“size,” units in parentheses, and dimensions L, W, D with separators 

(“x” marks). The corresponding data-body cells should accommodate the 

values for the three dimensions with the same separator marks “x.” 

The sample table in figure 18 uses pictures in the table data body to 

describe the attitude variable. If the attitude corresponds to the configuration shown in the picture, then the Cd value in the fourth column is the 

result. The interpretation of the rows is rich in meaning. The first row 

element develops seven multiples in the second column, but the seven 

multiple columns collapse into one element again in the third column. 

Again the fourth column has seven multiples corresponding to the seven 

items in the second column. The other rows also develop multiples and 

collapses. 

Figure 19 shows the very popular use of graphical icons or symbols, as 

used in the Co~~unzer Reports magazine. Each icon is explained in a footnote, end note, or other note (such as a key). 

Table 33. Dimensions of American Standard Companion Flanges* (ASA) 

(bll dimcruians in incha) 

160 lb 300 lb 400 lb GM) lb 

Nom pipe aim 

WXJ lb 1.5M) lb 

s Y 2 x Y z x 

-_---_ -I-----~Q--P-Fx Y z 

YIZ x Y 

z 

es % % IS w 

1% % % I; 

For aizes below 4 in.. 1% H I I H For aizca b&w 3 USC diuwmiom of 

I% 

1% I 

IX 

I 

i; 

II%6 ‘He ‘Hs 

in.. MC dimenIHs l%G 

1% 1% rz 

29fG l!iG ‘%a :H” 

the GM) lb !lnngcs INS 1x0 hxu of 1.500 lb 

2xs % N ZN I$y 

IHs . . . . . . . . . . . . t; 

2Hs 1% 1% 

l?is 

nvlces 1% 1% 

In:% 

. . . . . . . . . . 

:H 3KY,1fs I I% 

2% )Z I:, . . . . . . . . . 

19fs 

I% 

I!4 3x6 3’%s 

1% 

. . . . . . . . . . . . . . . . . 3x0 Ms IX0 . . . . . 

g 1% 

3 4!< I”<a l!iS 

31x6 I% 1% . . 

4% I’Ha I’Ka 

4% ::: 

. . . . . . . . . . . 

3H 4’Xr I ‘4 I!; 4% 

:E ;z 

I% 

IWS I’?<0 5 2% 2!4 5% 2% ;I 

. . . . . . 

: 

5%. 1%. IXS 

3% 

1% 

6Xs 1x6 

5% 2 

I’%, I’?<‘ 

IX6 

:H 

:Ls 

6 2% 

2 7 2% 

2K b!; 2% 2% 6% 3x0 3Ns 

: 

7:ic 1x0 1x6 

7x6 

9’Hs 

2Ks :K. 

1% 2% 7% 3% 3% 7% 4M 4% 

I% 

6% 2.L; 

1% IOH 

6% 

2x6 2x6 

2% 2% 9s 3% 3% 9 

1: I:, I 1x6 I’?46 

IOH 2’Hs :flNe 

12% 2% 

IOX 

4’Hs 

3 

41x4 

lI$i 4 

3% 12% 2% 4 :‘/6 

11% 

13% 

5% 

4% 14% 4% :% 

5% 

2!Cc 2”is 

I4 O.D. 

14% 2% 4 

15% 2,s 

14% 3% 

3!C 

4% 

16% 

15% 

14% 6!1 7 

3% 4% I6Y 4?i 5% 17% 7% 8% 

lb O.D. Ill 

::r. 

3ya I9 :x 

4% 

4% 

;pc 3%a 4% 17 I,$< 5% 6% 1% . . . . 

18 O.D. 19% 3’Ifo 21 3’Hs 5 19% 20 

20 O.D. & :z 4,x0 13% 3% 

3% 

2:: 

:;p :K 

21 3% 5% 

I!.; 6% 21% .._.. 

21% 

1:: 

4% 6 22!; 6 7% 23% . . . 

24 O.D. 

4% 

23% 4 

IO% 

27% 

5% 24 

4?<s 6 

6X 

27% 

:w 24% 6!5 6% 25% . . . . . 

4% 6% 26X 

11% 

7% 29% a IO% 30 . . . . . 13 

*Other dirncnaiom arc gircn in T.xbiea 31 and 32. Finiilrcd bore on lnplmd flnngc to bo such IIS method of ntcoclmcnt of pipe rcquirca. 

L 

Source: T. Baumeister and L. S. Marks, eds. (1964). Mechanical Engimws Hamibook, 6th ed., McGrawHill Book Co., New York, NY. 

Figure 16. A sample table connecting values to dimensions in a sketch. 

22 
Figure 17. A sample 

table showing highdensity polyethylene 

tank data having 

three size values in 

one column. 

Nalgened Sturdy Rectangular 

Tanks With Covers VERY POPULAR 

HIGH DENSITY POLYETHYLENE 

Qctan 

:ance. B 

ular tanks have excellent chemical, abrasion, and impact resishey feature continuous operating service lo 150°F. Large tanks 

Teed additional support at room temperature. 

Part No. Tank 

Tank 

“sl”,’ 

Size (In.) A proximale Wall Wilh Cover 

With Cover ( . LXWXO &ickness (In.) Price Each 

; ~~~~:~ : 8x8~8 14 x 10 x 10 % 

z 

45.50 502

19017LG 7 12X12X12 57.80 

1902OLG 1: 18 x 12 x 12 K2 78.00 

19022LG 24x12~12 552 170.00 

IE% z 

24x24~12 

; 

173.00 

299.25 

19024LG 6 zFf:2 

1::2’;:: 1: 12X12X18 18x12~18 

vz 93.10 

106.40 98.60 

1 zo”:;:: :: 24 18x 182 x 18 E K3 161.50599 

1 z:::: 63: 36 24x18~18 x 20 x 20 ‘xr XI- 227.50 135

19039LG 

:z 

18x12~24 159.90 

19041LG 24x12~24 169.60 

19044LG 24x18~24 ; 191.90 

19046LG 

:: 

36 x 24 x 24 Kr 325.50 

19047LG 94 

1EE :z 

sEF3~4 

51 391.90 

24x8~30 ; 

156.90 

173.70 

19051 LG ‘:A 30 x 30 x 30 x6 423.90 

19053LG 24x24~36 x5 

19055LG 140 30x30~36 Kr Ei 

liscount: Less 10% 2-4; 15% 5-7; 20% 8 or more. 

Source: Consolidated Plastics Company Inc., 1996. 

23 
Figure 18. A sample 

table containing 

pictures and 

collapsing branched 

rows. 

24 

Table 4. Drag Coefficients 

Object Attitude j CD 

-! / 

I-- 

: I 1.17 1.16 

Rectangular plate, 8 

sides a and b 0 = 12.5 j 

:i ’ 

1 

m / 

I 

L,fy, _I 1 /!i 

0 

, 

1 ! 

1 

2:oo 

Two distance diaks. I spaced apart 3 1.5 I 1 - = 2 / 

d 31 

& 1 1 @ j ;I!; 

ItI- I 

‘I v 0.91 

Cylinder 12:- 

0 %D 1 

0.85 

-= 4 7! d 0.87 

+~I-- 0.99 

Circular disk 

Hemispherical cup. 

oven back 

Hemisphericnl cup. 

open front. 

parachute 

Cone. closed base 

Source: T. Baumeister and L. S. Marks, eds. (1964). Mechanical 

Eq+ws’ Hancibook, 6th ed., McGraw-Hill Book Co., New York, NY. 
Figure 19. A sample 

table containing 

graphical icons. 

Qpping strategy 1 

You can buy all the camcorder 

corders wei h two to three pounds. FuIl-sized 

you need for SGOO. Use our 

W-IS mode f s wclgh . at lcast five pounds. 

Ratings to find the best picture Decide what to spend I’l;Cc ~n~tlcls smt nt 

quality and features. 3001it %SOO; liiglA& ~notlcls, at about %SOO. 

Know where to shop Electronics stores 

Know the choices The basic compact tape have rbe widest selection. Discount stores usufoments are 8mm and VHS-C. At standard ally have less selection but good prices. 

speed, 8mm models record for up to 2% hours; Departmerit stores carry few camcorders. 

VHS-C models, for up to 40 minutes. There 

arc “high band” versions of both forman (Hi8 Using the Ratings We tested 8111111, Hi8, 

and S-VHS-C), with enhanced video rcsolu- and VHS-C nmcordcrs. If you can’t find 

tion that’s displavctl at its best through a TV a wed model, call the manufacturer; see 

.set with an S-video input Compact cam- page 64. 

tYal1 Ratings L&cd in order of ooernl~scorc 

Source: Conszrmcr Rqmrfs, 1996. 

25 
Figure 20. Sample 

tables where 

elements can contain 

values of 

computations. 

Figure 20 shows sample tables in which elements can contain relationships in terms of other table values and inputs for interactive computations. These representations were the forerunners of work sheets and 

spreadsheets. By developing suitable representations for the elements in 

these tables, we can realize dynamic tables. Section 5 presented a discussion of tables with cell values computed dynamically as in spreadsheets. 

It is possible to continue this survey and discover many other table 

features in vogue in books. However, I believe I have cited a sufficient 

number and variety of examples so that we can discern the basic characteristics needed for computer representation of a variety of tables. 

C:H+.... 1 0.11 30.0; i 

1 

1.21 0.233 0.397 0.0926 0.325 0.07% 

CH,. . . . 0.95 16.04 13.63 0.X 0.593 0.4459 O.-I51 

N:. . . . . 0.01 SO” / o.%? 0.015 0.%-i o.oux 0.173 i 

0.3392 

0.0026 

Total . . 1.00 M = IS.12 l.WO c, = 0.5+22 c,. = 0.11x 

MC, at 60°F 

k= 8.985 _ 8.983 = 

8.985 - 1.99 6.993 

1_284 

Source: Hans Gartmann, ed. (1970). De Lazul Enginechg Hmdbook, 3rd ed., 

McGraw-Hill Book Co., New York, NY. 
7. Anatomy and Morphology of Tables 

The previous section’s survey of practical tables found in books, newspapers, and brochures suggests a minimal structure for adopting such tables 

for computer applications. The anatomy of a minimal table structure is 

shown in figure 21. The structure consists of a table caption, a table 

column header (and/or row header), a table data body, footnotes, and 

table data in various memory locations. The column header and data 

body consist of cell cages. The cell cages contain pointers to cells. These 

cells are of different types (sect. 7.2). The cell cages can take any shape, 

rectangular or nonrectangular, the latter accommodating multiples and 

collapsing multiples. This representation separates the table structure 

from its data. Thus, it is possible to define generic table types and operations on them. Specific domain data can be connected to table cells by 

Caption B Table 7-l. Thermal properties.* 

Header -44 

Cell cage*_1 1 

Data body * 

Cell cage k 

Footnote _k *Applicable to real gases. I Pressure psia 100 212 5.4 

Temperature deg F 150 239 4.3 

Contents in Volume ft3 200 320 3.6 

memory 250 360 3.1 

300 410 2.7 

Figure 21. A generic structure for a simple table. 

27 
Cell cage 

\ 

7.1 

pointers. Since table data are not in an array, there are no restrictions on 

their type. 

Cell Cages 

The cell cage (see fig. 22) consists of a number of cells, each of which is 

identified by row and column positions. In the example, 18 cells are 

arranged in a rectangular fashion (3 columns and 6 rows). Each cell is 

identified by the position it occupies in the cell cage (0,O to 5,2). The cells 

are themselves shown separately, and each cell can be of a specific type; in 

other words, they need not all be of the same type. By this arrangement, 

we can identify the relative position of the cells by the cell cage, while the 

cell type determines what can be stored in the cell. Pointers from a cell to 

the contents in memory will allow basic data types to be stored. 

Contents + 

inmemory 

/ 100 150 200 250 300 / 

1212 239 320 360 410 1 

15.4 4.3 3.6 3.1 2.7 , 

Figure 22. Cell cage and cells. 

28 

Cells 

“Fqz ------ -I 

L7 ,, 

5,2 
In a simple rectangular cell cage, adjacent cells bear certain relationships. 

If we take a cell i, j, its neighbors are north: i - 1, j; east: i, j + 1; south: i + 1, 

j; west: i, j - 1. For a cell cage with nz rows and n columns, i will be between 0 and nz - 1, and j will be between 0 and n - 1. Although the cell 

cage is rectangular, a single vector can be used to represent the cells if we 

arrange the cell cage position numbers in a sequence (a vector), as shown 

in figure 23. The position of a cell can be obtained from its sequence or 

serial number (Ns) by the following relationship: 

i = Ns/n and 

j = Ns % n. 

The sequence number (in the vector) for a given cell i, j is obtained from 

the equation 

N, = ilz + j. 

In terms of C++ programming language, a vector of void pointers corresponding to each N, value can point to individual cells. Depending on 

the cell type, its contents can be processed. As an example, figure 23 

shows integer data in a cell; the contents can be accessed via an integer 

pointer. 

Sequence 

number 

+o 1 2 3 . . . . . . 16 17 

pk.+ Cell type -k e.g., integer 

Integer pointer 

Contents* ilO 150 200 25 0 300 

212 239 320 360 410 / 

15.4 4.3 3.6 3.1 2.7 ~ 

Figure 23. Cell-cage sequence numbers. 

29 
We also need an arrangement for a nonrectangular cell cage representing 

a table that has a branching column. To demonstrate this case, I modified 

the table of figure 21 and present it as figure 24. 

In this case, it is immediately obvious that the table data body is still 

rectangular and can be represented by a rectangular cell cage. However, 

the column header should be represented by a nonrectangular cell cage, 

as shown in figure 25. Since we know that there are four columns at the 

junction of the column header and data body, the cell cage will have four 

columns, as shown in this figure. We choose to omit cell 0,2 in the first 

row of the cell cage. In general, we will follow a convention of missing the 

later cell number(s). We can also imagine that cell 0,l will stand for (the 

missing) cell 0,2, if ever we look for it. For example, if we look for the 

northerly neighbor of cell 1,2 by the conventional (previously described) 

procedure, we will get the cell 0,2, and by this just-stated rule, we will 

convert it to OJ. It may be worthwhile to note that the cell 0,l will have 

one more neighbor now than it would ordinarily have; it has two southerly neighbors. 

Figure 26 shows the vector representing the cell cage. It has seven sequence numbers connected to appropriate cell positions and memory 

locations. 

While the representation and the theory for cell cages with expanding 

columns and collapsing multiple columns are complex, algebraic relationships can be developed, programmed once, and then be made available 

for use by downstream users. For brevity, I do not go into further detail in 

this report. 

Caption .& Table 7-1. Thermal properties.* 

Header k Pressure 

psia 

Temperature 

deg F deg C 

Volume 

ft3 

Body k 

100 212 100 5.4 

150 239 115 4.3 

200 320 160 3.6 

250 360 182 3.1 

300 410 210 2.7 

Footnote -+ *Applicable to real gases. 

Figure 24. An example table showing a column with branches. 

30 
Cell cage 

Contents 

Pressure 

psia 

Temperature 

deg F deg C 

Volume 

fts 

Sequence 

number +r 

0 

Position in 

cell cage 

+c- EII 090 

0 

-I {’ -------- L, 

I 

193 

Figure 25. Representation of column header cell cage for table in figure 24. 

L- O,zL Note the missing cell 

Contents 

Pressure Temperature 

psia deg F deg C 

Volume 

ft3 

Figure 26. Cell-cage sequence numbers for a column header with branches. 

31 
7.2 Cells 

As we have seen, there are different types of cells. The simplest cells are 

those containing basic data types available in the programming language, 

e.g., char, integer, real. By defining other data types, we can easily extend 

the cells to contain real numbers and strings. With object-oriented programming, it is only a little more compJex to define cells of other types. 

Figure 27 shows a number of cell types and the data they contain. 

7.3 Table Types 

A spectrum of tables can be developed based on the parameters identified 

so far (see fig. 28). By using the representations described so far, software 

developers can generate several table types. First, a number of table types 

will be developed and made available as part of a Natural Computing 

tool box. In an iterative cycle (see fig. 29), developers will address almost 

all the requirements over several versions and years. Domain specialists 

will incorporate their knowledge into these tables and validate their 

correct and appropriate operation. Domain specialists will call for new 

versions to be developed by software developers if the existing table 

representations are inadequate in certain features. End users will use the 

tables provided to them by the domain specialists. Specialists will incorporate new domain knowledge, catering to the needs of end users. 

Cell type Data type Example 

r~ Numeric I k Number 2.34 

FI -+ String Pressure 

nw String or char - 

I I 

I Parenthetic I---+ Two of type 212 (100) 

len;ni , Graphic or icon m 

llluitifrng , Multiple strings Pressure, ps,ia 

I Units string I -k String(s) (from a list) Btu per hr ft* 

I Composite string w I 

Combination Composition (balance Al) 

Figure 27. Example cell types with sample data. 

32 
Header 

_ Column 

header 

_ Row 

header 

- Both 

Cell cage Cell 

.~ 

- Rectangular - Numeric 

.~~~ .~ .~ _ _ ~~ 

- Multiples I String 

- Collapsing I Blank 

multiples 

- Combination - Graphic 

- Composite 

interpolation 

I Yes 

I No 

Figure 28. A spectrum of tables in profile representation that can be generated. 

Notes 

- Footnotes 

- Other notes 

Representation Incorporation Utilization 

New version New domain 

Figure 29. Iterative cycle of table development in future Natural Computing system. 

Two types of system growth can be anticipated. One is the domain 

growth and the other is the Natural Computing tool growth. Initially, a 

set of Natural Computing tools will be developed and presented to 

domain specialists for use in their domains. A variety of domains can be 

incorporated. As more complex domain problems are incorporated, new 

Natural Computing tools will need to be developed every time a certain 

new table type is encountered for which a table representation is not yet 

available. As new table features (structures or behaviors) are encountered 

or invented, software developers will play a primary role in developing 

table objects with needed features. Once those features are implemented, 

the newer versions of Natural Computing software tools will become 

available to all domains. Eventually, most types of tables will be represented in Natural Computing and any domain can be captured. 

33 
8. Computer Representation of Tables 

The foregoing discussion suggests that representing tables as objects with 

various attributes and methods for operating on their contents is promising. It is then obvious that object-oriented programming techniques and 

languages will be most suitable and perhaps essential in such a representation. A table class will be developed comprising a base class and a 

number of derived classes. As the need for new table features is recognized, more derived classes can be developed and added to the system. A 

container class should be chosen for a table class since it contains several 

components as identified in section 7. The component classes will include 

a caption, a header, a data body, and various notes classes (e.g., footnotes). The header and data body classes will consist of cell cages and 

arrays of cells. The cells themselves are represented in terms of a base 

class and a number of derived classes. The table hierarchy is shown in 

figure 30. (I reserve low-level details for a later report, as including them 

here would hinder the smooth presentation of the subject matter.) 

Data are entered into a table at the cell level, the table being built up from 

the components by the various constructors in the table class. A number 

of operations must be defined to manipulate the contents of the table. 

Creation, deletion, modification, storage, recall from storage, and 

persistence operations are performed by conventional methods. Display, 

printing, and plotting of tables (as they appear in books and documents) 

(U.S. GPO, 1984) will be another set of operations. Perhaps the methods 

available up to this point in current and emerging state-of-the-art text 

processing systems will suffice for the display functionality of tables. 

Additionally, the table class will have operations that make the table serve 

its functional roles (for example, the relationship between input and 

output). 

As stated in the introduction, the main role of a table is to serve as a 

functional relationship. That is, given an input value (or values), the table 

object should provide a result value. This functionality should be available both as an interactive feature and as part of a procedure (or program) 

building capability. In executing this function, the table class should 

behave responsibly by properly interpreting the qualifying information 

Table Caption 

base Header Cell 

class Data body > base 

F 

Notes class 

I, 

Y 

, I 

Figure 30. A hierarchy of table classes and component classes. 

34 
available in the header(s), footnotes, and other notes. Thus a table is not 

merely a lookup table; it both looks up information and looks out for 

footnoted interpretations based on conditions. I call this the responsibility 

of a table object during computations. 

In programming, a table will be even more useful and helpful to a naive 

end user if it can adjust units automatically, Every quantity in a table 

should know that it carries units. In general, all calculation features in 

Natural Computing work with a set of four items. These four items are 

(1) a name for the characteristic, (2) a variable representing the characteristic (called notation), (3) the value of the characteristic, and (4) the units 

of the characteristic. The characteristic belongs to or is an attribute of a 

system under discussion or calculation. A variable is a symbol chosen to 

represent the physical quantity or characteristic for brevity in writing 

relationships, equations, or table header column cells. The characteristic 

has a definite value at a given state. Its value is given in some physical 

units. (In many documents, the relationships between the variables and 

their descriptions are often given in lists of notation or lists of symbols.) 

A table header may contain a variable and its description and should be 

connected to the corresponding pair in the list of notation. The units of a 

quantity are given both in the notation and in the table and should be 

compatible but not necessarily identical. For example, a quantity such as 

velocity can be in feet per second and in miles per hour. When a table is 

queried, the units in the query (input) and the units for the quantity in the 

table should be compatible. Likewise, the units of a table quantity should 

be compatible with the units of the output quantity. The variable and 

units pair will also be connected when procedures are developed that use 

variables from tables. By treating units (and conversions) simultaneously 

with functional manipulation, we avoid the units-related errors that have 

caused much grief in programs of yesterday. 

Other sets of operations exist that make tables very effective in programming: for example, mapping a table of one type into another. People 

usually copy tables of one type into another type. This can be done in 

software by the development of mapping operations that change table 

types and permit the development of supersets and subsets. These operations will enable abstracting part of a table data body and creating a 

subset table. In reverse, two tables can be combined to form a superset. 

For example, a table of Year versus Income and one of Year versus Expenditure can be combined and a superset table defined with three 

columns: Year, Income, and Expenditure (see fig. 31). We could further 

extend this table to yet another superset table by defining another column 

called Profit, which may be defined as the difference in the Income 

column and the Expenditure column. Thus, tables should be able to 

generate new tables by addition and/or deletion of columns and rows. 

Mathematical operations on tables make them very valuable and highly 

useful. For example, figure 32 shows how two table values can be added 

to develop a result table. We should be able to achieve this result by 

35 
Figure 31. 

Development of a 

superset table from 

two tables. 

simply writing a functional relationship, such as result table = table 1 + 

table 2. Other operations to be incorporated in tables are subtraction, 

scalar multiplication, division, columnar multiplication and division, and 

other complex operations. 

1960 132 1960 126 

1970 130 1970 115 

1980 136 1980 124 

1990 145 1990 137 

Year Income ($M) Expenditure ($M) 

1950 143 110 

1960 157 126 

1970 156 115 

1990 1980 I 1 16684 12437 I 

Income for North Region Income for South Region Income for North & South Regions 

Figure 32. Development of mathematical operations with tables. 

36 
9. Choice and Use of Tables 

As shown in the foregoing section, tables come in a number of different 

types. Just as word processors contain document manipulation tools and 

graphics packages contain picture manipulation tools, a Natural Computing software tool box will contain several table tools for use by domain 

specialists. A domain specialist can select a table that best fits the needs of 

a particular application. Thus, an instance of a blank table is created 

(instance creation is the responsibility of the table object’s constructor). In 

the creating/editing mode, all data and information are entered into a 

table. It is the domain specialist’s responsibility to fill in the footnotes and 

other notes as applicable. The table object also develops a number of 

behavioral characteristics for the domain specialists’ review. For example, 

the limit values (minimum, maximum, and singularities) of a characteristic are recorded, and the domain specialists check them for accuracy and 

applicability The table object uses these limits to flag an error message 

when a user tries unallowable values; this is another feature imitated 

from human usage of tables. In real-world applications, domain specialists often provide such checks. Minimum and maximum values of a 

variable prevent extrapolation outside allowable bounds. 

A table object will also have a set of input (or query) templates and 

output templates. Recall that a table is a functional relationship between 

quantities (characteristics) identified in the cells of the column header. 

Given one of the quantities, all others can be determined from the table. 

We can use this functional relationship in generating query templates. 

The user can choose the appropriate input template, type in a value for 

the independent variable, and submit it to the table object. The template 

also guides the user with the limits on the variable values. These guidance values in the templates protect the table from invalid or out-of-range 

queries. A table will return a result by means of the output templates. A 

table with input and output templates can be likened to a hardware 

component with its input and output sockets. These input and output 

templates are very useful in connecting different functional objects into a 

procedure or program. 

37 
10. Search for Data 

Since the cells in a table can be connected by adjacency lists and pointers 

to neighbors, and pointers lead to data values or contents, searching a 

table for result data items is quite simple. A query consists of an independent variable/value pair and another dependent variable. If we wish to 

find the value corresponding to the dependent variable, we use the 

simple property of a table that two value cells bear the same neighborly 

or adjacency relationship as the corresponding two variable cells do. This 

relationship is depicted in figure 33. 

Figure 33. Adjacency 

relationships are used 

to get table values. 

38 

Question: 

Given pressure = 200 psia 

Find volume = ? ft3 

Table 7-l. Thermal properties.* 

_____________e, 

r_____________-, 

I I 

I I 

: Pressure ! Temperature 

I 

i Volume ; 

I I psia i I I 

f I 

deg F ! ft3 I I I I ‘__ ____________ .____.._________1 

100 212 5.4 

150 239 4.3 r-- -------- r____------- --. I I I I 1 

I I 200 i I 320 1 i 3.6 

I I 

I 

I 1 

___________d ____________ .__a 

250 360 3.1 

300 410 2.7 

I I 

I 

I 

I 

1 

*Applicable to real gases. 

Solution: Adjacency relationships: 

Pressure + second easterly neighbor -+ volume 

Hence, second easterly neighborof 200 is + 3.6 
11. Visibility of Table Data 

An advantage of the use of tables in Natural Computing is that they 

provide the user a way to gain insight into the computation. Educators 

criticize current computational software systems as “black boxes,” because the solution method is incomprehensible from the software. The 

user learns nothing from using the software. For that matter, even domain 

specialists do not understand what is in the code, once their domain 

information (actually, algorithms shorn of all information) is put into code 

by a software developer. With traditional media (paper, calculator, and 

pencil), students’ learning improves proportionally with the number of 

problems solved. In contrast, with current computer software, a student’s 

learning does not improve with the number of problems solved. Since 

tables in Natural Computing, as described here, reveal themselves and 

show relationships between variables, a student can realize opportunities 

available and watch out for pitfalls in the problem domain represented by 

each table. 

39 
12. Testing a Table in Isolation 

Natural computing tables have the advantage that they can be tested in 

isolation. Testing is a key task in software development; capturing an 

application domain in software is equally critical. Isolating a table and 

testing it for a variety of inputs, together with the built-in justifications, 

limits, behaviors, and responsibilities prescribed for a table class, would 

go a long way toward eliminating bugs in software that used tables as 

described in this paper. Since the filters on a table will allow only 

preapproved types and ranges of values, isolated testing can come very 

near to guaranteeing both the software and the domain knowledge (the 

application). 

40 
13. Embedding Tables into Text 

In textbooks and other paper documents, tables are embedded in text. A 

reader can read them and get a general idea of the information presented. 

Or a reader can interactively use the table by looking for values, with 

some input parameters. Any minimal computer representation should be 

able to duplicate that capability. 

In Natural Computing, tables are embedded in text (as in fig. 34) and 

when a user wishes to use tables interactively, the tables can be activated 

by a computer command. Input and output templates (boxes) will appear, 

allowing the user to type in the input values and receive output values. 

Such an interactive facility is useful for studying a table and understanding trends in the domain. 

While developing a procedure or a program, a user can copy and paste a 

table from the text. In this case, the table object is available to the procedure; this procedure can be saved separately from the original text object. 

In software as in paper-based computation, people should be able to 

develop procedures by connecting several tables, equations, and graphs 

to inputs and outputs. 

41 
22.2 H&R Block Basic Income Tax Course 

PRACTICE 22.2. Prepare Form 1040X to correct Charles Albertson’s 1985 return which was filed April 

10, 1986. On May 10, 1988, he received another W-Z for 1985. The W-Z had been lost in the mail. 

Federal Tax Wages 

$300.00 $1,700.00 

FICA 

$119.85 

State Tax 

$110.00 

Employee 

C.. Albertson 

While talking to Mr. Albertson, you learn that he paid all of the cost of maintenance of his home, which 

was the principal residence of his son, Johnnie (201, whose gross income was $600 for 1985. Assume that 

you have completed a support worksheet and that Mr. Albertson provided $3,000 of Johnnie’s $3,600 total 

support. 

In column A, on the Form 1040X, we have entered the information needed from Mr. Albertson’s original 

Form 1040. The original return has not been examined and he has not been notified that it will be examined. 

Notice that line 4 (deductions) of Mr. Albertson’s Form 1040X is blank. That’s because in 1985 the standard 

deduction, otherwise known as the zero bracket amount, was built into the tax table. Thus, taxpayers not 

itemizing deductions did not need to claim a standard deduction on the tax form. 

The necessary portion ol the 1985 Tax Table is shown below. 

14.000 

14.000 14.050 1.766 1.391 2.109 X.686 

14.050 14.100 1.776 1.399 2.122 1.695 

14.100 14.150 1.786 1.407 2.134 1.704 

14.150 14.200 1.796 1.415 2.147 1.713 

Id.200 14.250 1.806 1.423 2.159 1.722 

14.250 14.300 1.816 1.431 2.172 1.731 

14.300 14.350 1.826 1.439 2.16d 1.740 

14.350 14.400 1.836 1.447 2.197 1.749 

14.400 14.450 1.846 1.455 2.209 1.756 

14.450 14.500 1.856 1.463 2.222 1.767 

14.500 14.550 1.866 1.471 2.234 1.776 

14.550 14.600 1.876 1.479 2.247 1.785 

14.600 14.650 1.886 1.4i-I 2.259 1.794 

14.650 14.700 1.896 1.495 2.272 I.803 

lA.700 14.750 1.906 1.503 2.284 1.812 

14.750 14.800 1.916 1.511 2.297 1.821 

14.8M) 14.850 1.926 1.519 2.309 1.830 

14.850 14.900 1.936 1.527 2.322 1.839 

14.900 14.950 1.946 1.535 2.334 1.846 

14.950 15.000 1.956 1.543 2.347 1.857 

Source: H&R Block (1990). Tus Prqmvfion Gzk’es. 

Figure 34. Example of a table embedded in text. 

42 
14. Tables and Databases 

Although databases are commonly thought (albeit mistakenly) to represent tables, the foregoing description of tables should counter that impression. A brief comparison of database systems and real-world tables 

shows how much more complex actual tables are than the “tables” that 

database people claim to represent. 

The vast literature on database systems might lead an unwary reader to 

conclude that the database community has already represented tables in a 

computer-usable format. However, after calling a table a relation (Date, 

1995, p 79), Date then states, “a relation and a table are not really the same 

thing, although in practice it is frequently convenient to pretend that they 

are” (p SO). In a discussion of formalizing the concept of a table, Shaler 

and Mellor (1988) state that normalization rules can be viewed from two 

perspectives. The first focuses on the form of data in databases. The rules 

tell how to set up tables so that little redundancy is in the data; that is, the 

amount of data required to store a certain information content is minimized. The second perspective (the one most nattlval to us) looks at the 

normalization rules as statements about the repertoire of forms that we 

use in our model (the fact that we are using tables, for example) and at 

the meaning we imply whenever we use a form in a particular manner. 

Date (1995) describes a database system as basically a computerized 

“record-keeping” system. Its overall purpose is to maintain information 

in a computer and to make that information available on demand to three 

classes of users: applications programmers, end users, and database 

administrators. In a slightly more precise definition, Date says, “A database consists of some collection of persistent data that is used by application systems of some given enterprise.” 

In defining a relational system, Date states that data in a relational system 

are perceived by the user as tables (and nothing but tables) (p 22). He 

goes on to state, “For most practical purposes, indeed, the terms relation 

and table can be taken to be synonymous.” The relational model is a way 

of looking at data-that is, it is a prescription for a way of representing 

data (namely, by means of tables). 

Continuing their discussion of regular tables, Shaler and Mellor explain 

the rules used in databases: 

First Rule: One instance of an object has exactly one value for each 

attribute. 

There is one and only one data element at each row-column intersection. 

This rule forbids the “repeating group” construct found in some databases (table 6) and true holes (as in table 7). 

Second Rule: Attributes must contain no internal structure. 

This is another expression of the requirement for fully factored attributes. 

The rule forbids tables with an internal structure such as table 8. But it 

allows a construct such as table 9. 

43 
Table 6. Example 

of a table with a 

forbidden repeating 

group. 

Table 7. Example 

of a table with a 

forbidden true hole. 

Table 8. Example 

of a table containing 

forbidden internal 

structure. 

Table 9. Example 

of a table with an 

allowable structure 

to represent 

information in 

table 8. 

44 

Owner Name of 

pet(s) 

Address 

Smith Rover 

Rin Tin Tin 

Sarzak 

100 Canine Court 

Jones Lassie 6 Dogwood Lane 

Owner 

Brown 

Green 

Jones 

Model 

Sedan 

Van 

Truck 

Manufacturer 

Ford 

Chevrolet 

License No. 

16923A 

23004C 

29-A-101 

Name 

Lassie 

Laddie 

Fifi 

Sex/breed 

F-Collie 

M-Collie 

F-Poodle 

Name Sex 

Lassie F 

Laddie M 

Fifi F 

Breed 

Collie 

Collie 

Poodle 

These examples illustrate that database relations do not allow the representation of the tables that scientists, engineers, and analysts use in a 

variety of domains, as described in section 6. This condition is due to the 

mathematical rigor of the database systems. Software developers are 

caught in a dilemma between mathematical rigor and the natural but 

highly flexible forms people use. 

Study the baseball tables in figures 4 and 35. Young children grasp the 

nuances of these tables. These simple-looking tables represent many 

relationships. Lay readers can quickly compare and calculate desired 

outcomes. As new games are played each day, new tables of values can be 

calculated from old table values and predefined relationships (formulas). 

The simplicity of the natural tables in representing a variety of relationships is yet to be matched by the best database systems. For example, I 

show in section 6 that ranges are represented, blanks are allowed with 

definite meanings, and data types are mixed with no problems. Notice 

how a batter and information on his field position are combined in the 

first column of the table in figure 35. 
Figure 35. A table of NEWYORK AB R H BI BB SO AVG 

baseball results 

representing a player Boggs 3b 4 0 1 1 0 0 0.331 

and his position in Girardi c 40 000 0 0.282 

same element. O’Neill rf 301010 0.368 

TMartinez lb 4 0 0 0 0 0 0.246 

Sierra dh 323010 0.281 

Duncan 3b 40 000 0 0.343 

GeWilliams If 3 0 0 1 0 0 0.333 

RRivera cf 2 0 1 1 1 0 0.5 

BeWilliams ph 1 0 0 0 0 1 0.278 

DJeter ss 311000 0.275 

TOTALS 31 3 7 3 3 1 - 

Why did I choose the baseball example? A six-year-old understands a 

baseball table. A child of four understands a table of menus and prices at 

a favorite restaurant! The knowledge of how to process tables is general. 

Humans gain such process knowledge independent of the domain. 

Initially, they apply the processing knowledge to simple (favorite) domains and later extend it to complex domains. A student may have 

problems with school homework but understands tables comparing 

automobiles in the April issue of Co~~nzcv Reports. These tables do not 

even use numbers in the cells. Circles filled with red and black colors are 

used (see fig. 19), and footnotes explain what the various colored circles 

mean. As people grow, they comprehend more complex table constructions. With help, they understand tables of increasing complexity. And 

they continually add more complex tables to their repertoire. That is the 

nature of our learning. 

Although I argued in the foregoing that databases are not representations 

for tables, it is by no means suggested that tables (even Natural Computing tables) will replace databases. Natural Computing will provide 

interfaces to databases so that their attractive features can be used in 

addition to tables. Such a strategy allows tables to be used in their most 

natural form while large amounts of data can be handled efficiently 

through databases. 

45 
15. Example Usage of Natural Computing Tables 

Based on the examples given in the survey in section 6 and the scheme for 

representing tables in computers given in section 8, one will be able to 

produce documents containing a variety of tables. As stated previously, 

such tables will manifest themselves in three basic forms: (1) tables 

embedded in text, (2) tables that can be used interactively, and (3) tables 

that can be built into procedures. Figure 36 shows a table embedded in 

text, along with the menus available to manipulate both the document 

and the tables. When the user wishes to use the table to obtain values, he 

or she activates the table (changing it to an interactive mode) as shown in 

figure 37. In this case, a highlighted table appears and presents input and 

output boxes (not shown in the figure in order to avoid clutter). When the 

user types in an input value, result values are outputted by the system. As 

stated in section 12, this interactive mode is used to test the table in 

isolation. An end user uses this mode to obtain values to comprehend 

trends or to use the result values in a series of calculations of a temporary 

nature. Finally, when a domain specialist wishes to incorporate a table 

into a procedure, a procedure is invoked and a table is connected to it, as 

shown in figure 38. Once a table is set into a procedure, it is embedded 

inside the procedure, and calculations are completed with the entire 

procedure (fig. 39). 

46 
Figure 36. A table 

embedded in text 

with Natural 

Computing menus. 

Action Features Edit 

Show n Text Undo 

Interactive use Equations cut 

Program use Tables COPY 

Setup Spreadsheet strips Paste 

Options Graphs Clear 

line. The equation of this line can be Unit strips 

the strengths and wire diameters. Th 

L=$ (8-10) 

where A is a constant related to a strength intercept, and m is the slope of the line 

on the log-log plot. Of course such an equation is only valid for a limited range of 

wire sizes. Table 8-2 gives values of m and the constant A for both English and SI 

units for the materials listed in Table 8-1. 

Although the torsional yield strength is needed to design springs, surprisingly, very little information on this property is available. Using an approximate relationship between yield strength and ultimate strength in tension, 

s, = 0.754, (8-11) 

and then applying the distortion-energy theory gives 

s,, = 0.577s, (8-12) 

and provides us with a means of estimating the torsional yield strength S,,. But 

this method should not be used if experimental data are available; if used, a 

generous factor of safety should be employed, especially for extension springs. 

because of the uncertainty involved. 

Variations in the wire diameter and in the coil diameter of the spring have an 

effect on the stress as well as on the spring scale. Large tolerances will result in 

* Sec. for cxamplc. the second cdtnon of thts book: Joseph E. Shlglcy, “Mechanical Engineering 

Design.” 2d cd.. p 362. McGraw-Hill Book Company. New York. 1972. 

Table 8-Z CONSTANTS FOR USE IN EQ. (g-10) TO ESTIMATE THE TENSlLE 

STRENGTH OF SELECTED SPRlNG STEELS 

Mncaial 

Conslull, A 

Size nnge. Exponen. - 

lnm ” kpsi MP. 

Music wire’ 

Oil-tempered wire’ 

Hard-drawn wire’ 

Chrome vanadium’ 

Chrome sihcone 

0.Ou40.250 0.10-6.5 0.146 196 2170 

0.020-0.500 oso--12 0.186 149 1880 

0.02%o.scKl 0.70-12 0.192 136 1750 

0.032-0.437 0.80-12 0.167 169 .2m 

0.063-0.375 1.6-10 0.112 202 2ooo 

47 
Action 

Show 

Features 

Text 

Interactive use 1 Eauations ~ ~, 

Program use 1 Tables 

Setup 3 -I 

Edit 

Undo 

cut I 

I COPY 

-Gadsheet strips Paste 

raphs Clear 

Figure 37. A table activated for interactive use. 

48 
Figure 38. A table 

being set into a 

procedure. 

Action 

Show 

Interactive use 

Program use 

Setup 

Options 

. . . 

Features 

Text 

Equations 

Tables 

Spreadsheet strips 

Graphs 

Edit 

Undo 

cut 

COPY 

Paste 

Clear 

T. P V. relationship for gas X 

Temperature Pressure Volume 

(deg F) (psia) (ft? 

100 235.4 2.36 

\ 150 34.7 2.15 

‘I200 46.91 2.01 

250 54.8 1.96 

Step k is given temperature; find pressure from table. 

49 
File: Helical SpringsACS 

0 1 Show Text Undo 

I b fntcrar?tivP r_tSS Tables cut 

Stresses in Helical Springs 

,e Equations copy 

Figure 1 shows a round-wire helical 

spring loaded by the axial force f. v\ 

Setup Graphs Paste 

the mean spring diameter and das t OptiOnS Unit strips Clear 

Now imagine that the spring is cut a 

portion of it removed, and the effect ... Work shet 

I I 

portion replaced by the internal forces. Then, the cut 

portion would exert a direct she 

torsion Ton the remaining part 

superposition, the maximum str 

computed using the equation 

I Tmax=+y+ 2 

Replacing the terms by 

T= FDl2. 

I I r= dl2, 

I I J=rd4/32,and 

A=nd2/4 

In this equation, the subscript inc 

shear stress has been omitted a 

unnecessary. The positive signs 

retained, and hence Eq. (b) give 

the inside fiber of the spring. 

Now define spring index 

as a measure of the coil cutvatu 

Eq. (b) can be arranged to give 

Figure 39. A Natural Computing screen showing a document with a procedure. 

50 
16. Conclusions 

Tables are ubiquitous means of depicting functional relationships used 

freely in paper-based documents, such as textbooks, handbooks, journals, 

newspapers, and flyers. Although tables are complex representations, 

people use them with consummate ease. Although database systems have 

been misunderstood to represent real tables, they are at best representations of simple tables containing discrete domain data. Although database 

systems have been and will continue to be highly useful, they are not 

replacements for true tables. I have demonstrated in this paper the need 

for a richer and more appropriate computer representation for tables. 

The development of a base table class and several derived classes is the 

appropriate method to represent and use tables. The structure of a table 

has been analyzed, and the components have been identified. The table 

anatomy presented in this report has the advantage that it allows a 

variety of tables to be represented and used. Various component classes 

are also described. Interpolation and extrapolation, where permissible, 

can be performed with the table methods briefly presented in this report. 

The display of information in natural-looking tables both allows the user 

to visualize opportunities in the data in a table and warns the user of 

pitfalls in the data. 

If tables are represented and used as described in this report, computing 

will take a giant step toward making a reality the natural forms of data 

representation (such as tables). Once such forms of representation and 

processing are available, we can realize electronic handbooks, textbooks, 

documents, journals, and bulletins with the capability to interchange data 

and objects seamlessly. Knowledge from these various electronic library 

sources can be exchanged and combined, and computational procedures 

and systems can be developed that will be more powerful, economical, 

and expeditious than those created in the traditional software development cycle. 

53 
Acknowledgments 

l 

52 

The author appreciates Ms. Laura Mann’s efforts in obtaining permission 

to use copyrighted material from several publishers. 

The author gratefully acknowledges permission to use copyrighted 

material granted byMcGraw-Hall Book Co., New York, NY, 

The Washington Post Company, Washington, DC, 

Consolidated Plastics Company, Inc., 

Consumer Reports, New York, NY, and 

H&R Block Company 
References 

Baumeister, T. and L. S. Marks, eds. (1964). Mechanical Engineers' Handbook, 6th 

ed., McGraw-Hill Book Co., New York, NY. 

Coad, Peter, and Edward Yourdon (1991). Object-Ovientecl Analysis, 2nd ed., 

Yourdon Press Computing Series. 

Cumpsty, N. A. (1989). Coqressor Aerodynamics, Longman Scientific and 

Technical, Harlow, England. 

Date, C. J. (1995). An Infroducfiorz to Database Systems, 6th ed., Addison-Wesley 

Publishing Company, Reading, MA. 

U.S. Government Printing Office (GPO) (1984). U.S. Governnzenf Printing Office 

Style Munuul (March). 

Gartmann, Hans, ed. (1970). De La& Engineering Hundbook, 3rd ed., McGrawHill Book Co., New York, NY, l-18. 

H&R Block (1990). Tax Preparufim Guides. 

Karamchetty, Som D. (1997). Natural Computing, U.S. Patent No. 5,680,557 

(October 21). 

Karamchetty, Som D. (2000). Nafzml Computing: Ifs Impact OII Software Development, U.S. Army Research Laboratory, ARL-TR-2040. 

Keenan, Joseph H., Frederick G. Keyes, Philip G. Hill, and Joan G. Moore 

(1969). Steam Tables: Tkermdynanzic Properties of Wafer hluditzg Vapor, 

Liquid, and Solid Phases, John Wiley and Sons, Inc., New York, NY. 

Lemay, Laura (1996). Teuck Yourself Web Publishing With HTML 3.0 in u Week, 2nd 

ed., Sams.Net Publishing, Indianapolis, IN, 194,440. 

Lester, C. B. (1994). Hydruzdics for Pipeliners, Vol. 2. Fundunzenfuls, 2nd ed., Gulf 

Publishing Co., Houston, TX. 

Microsoft Word Version 7.0 (1995). Microsoft Corporation, Redmond, WA. 

Morris, Mary E. S. (1996). HTML for Fun and Profif, Sunsoft Press. 

Shaler, Sally, and Stephen J. Mellor (1988). Object-Oriented Systems A~za~ysisModeling the World ilz Data, Yourdon Press Computing Series, Prentice 

Hall, Englewood Cliffs, NJ. 

Shigley, Joseph E. (1977). Meckarlical Engineering Design, 3rd ed., McGraw-Hill 

Book Co., New York, NY. 

Van Wylen, Gordon J., and Sonntag, Richard E. (1965). Fundanzenfuls of Classical 

Thermodynamics, John Wiley and Sons, New York, NY. 

Streeter, V. L. (1971). Fluid Meckaflics, 5th ed., McGraw-Hill Book Co., New 

York, NY. 

The Washington Post (1996). Monday, May 27, p C6. 

Wolfram, Stephen (1991). Mafkenrafica, A System for Doing Mathematics by Cornpufer, 2nd ed., Addison-Wesley Publishing Company, Reading, MA. 

53 
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13. ABSTRACT (!&wimom 200 words) 

Some fundamental objects in practical documents have not been implemented in software so that 

they can be used easily for calculating. One such object is the table-despite the mistaken view 

that databases are adequate representations of tables. A survey of practical tables found in a 

variety of real-world documents reveals that many of their useful features are not captured in 

software. This report proposes data structures and computer representation for table objects. 

Through the adoption of such structures and representations, practical table objects can be 

developed for use by domain specialists. Such tables embedded in electronic documents can be 

used in interactive applications to retrieve data, but most importantly, they can be used as 

functional representations for copying and pasting into procedures and programs. Use of these 

table objects, together with other natural computing objects (such as equations, graphs, and 

procedures), will permit electronic documents like handbooks, textbooks, journals, and bulletins 

to be used seamlessly for calculations by both domain specialists and naive users. Such 

developments will reduce the lag between the availability of information and its use in 

calculations, and encourage the further development of knowledge. Software development for 

computation will change and its costs will be contained. 

14. SUBJECT TERMS 

Natural Computing, software engineering, object-oriented programming 

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