# Philosophical Transactions of the Royal Society (1665-1886)/Volume 3/Number 33

Contents:
* Contents
* An Introduction to the Fourth Year of these Tracts.
* ⁠An Account of the Invention of Grinding Optick and Burning-Glasses of a Figure not Spherical, lately produced before the R. Society.
* Some Observations by way of Answer to some of the Inquiries concerning Tydes, proposed Numb. 17. and 18.
* Queries and Directions for the Caribbe-Islands.
* An Account of two Books:
** I. SAGGI DI NATURALI ESPERIENZE satte nell' ACADEMIA del CIMENTO, in FIREZENZE A. 1667. in Fol.
** II. VERA CIRCULI & HYPERBOLÆ QUADRATURA, in propria sua Proportionis specie inventa & demonstrata à JOCOBO GREGORIO Scoto, PATAVII in 4° 1667:
*** Whence a Method is deduced of Measuring the Area of an Ayperbole, and consequently finding the Logarithim of any Number, and é contra: And this applyed to the Gauging or Mensuration of any Segment of any Cone, &c.
Numb. 33 Beginning the Fourth Year.
PHILOSOPHICAL
TRANSACTIONS.
Monday, March 16. 166⁠7/8⁠
The Contents.
An Introduction
To the Fourth Year of these Tracts.
In my first Number (of March, in the Year 1665.) I render'd the reasons and purposes of these Philosophical Communications. In the Second Year (beginning in March A. 1666. at Numb. 10.) I endeavour'd to revive and impress the Utility of these United Correspondencies. In the Third year beginning in March A. 1667. ​at Num. 23.) I defended our Industry from the Obloquies of such men, as prefer endless Contentions about words before the useful Works of the noblest Arts; and boast the Notions, yea, and oft-times the Cavils of pore-blind Heathen Writers, above the great and admirable Works of God.
Here, I think, I may from manifest appearances, without any great presumption, ominate, That the following Tracts shall somewhat more satisfie the Ingenuous, than the former, for as much as my Philosophical Commerce from time to time enlargeth it self, and I am still better and better furnish'd with store of judicious Correspondents in the most considerable places of the World; and they are by the foregoing Tracts (especially from Numb. 11. and several of those that follow) better directed to afford true Aids, as also to send in good Answers to the Enquiries already made publick.
Yet I hope, even our former Tracts will not be very much blamed by such, as shall be pleas'd to consider, that some of them have already brought in several pertinent Answers; viz. from a Sea-Voyage, the Caribbe-Islands, and Jamaica in particular, the Baltick-Sea, our Mendip-Mines, &c. Numb. 19, 27, 30; that others of them do instruct, prepare, and enable, for severe Observations; others lay the ground for Philosophical Advancements; others are accurate Exemplifications; and some of them contain divers valuable Particulars, which perhaps had otherwise been lost, or drown'd in a worse crowd of Impertinencies, or scatter'd in more costly Volumes. Certainly there are well-devised Directions, especially proper for Sea-men and all Travellers, and such as may pleasingly and beneficially entertain them whilst they are under Sail, or on their Land-Voyages; and they cannot be unacceptable to the ingenious and Curious in our Colonies, and in other places of best note. All which may be seen in a short view in the Tables annexed to Numb. 22. and 32.
An Account
The ingenious and industrious Francis Smethwick Esquire, Fellow of the Royal Society, having for divers years painfully search'd after the way of grinding Glasses not-Spherical, affirms, that at length he hath now found it; for the proof of which, he lately (viz. February 27. 166⁠7/8⁠.) produced before the said Society certain Specimina of that Invention, which were a Telescope, a Reading and two Burning-Glasses.
The Telescope was about four foot long, furnish'd with four Glasses, whereof the three Ocular ones, Plano-convex, were of this newly-invented not-Spherical Figure, and the fourth a Spherical Object-glass. This being compared with a common, yet very good Telescope, longer than it by about four inches, and turned to several Objects, was found by those of the said Society that look'd through them both, to exceed the other in goodness, by taking in at greater Angle, and representing the Objects more exactly in their respective proportions, and enduring greater Aperture, free from Colours.
The Reading Glass of the same Figure being compared with a common Spherical-Glass, did far excell it, by magnifying the Letters, to which it was applied, up to the very edges, and by shewing them distinctly from one brim through the Center to the other, which the Spherical Glass came far short of. And this effect the new-figur'd Glass perform'd only on one of its sides, and not on the other, as being of a different figure from Spherical-Glasses, which perform their effect near-equally on both sides.
Lastly, The two Burning-Concaves of this new-invented figures were, the one of six inches Diameter, its focus three inches distant from the Center thereof; the other of the same Diameter, but less concave, and its focus 10 inches distant. These, when approach'd to a large Candle lighted, did somewhat warm the faces of those, that were 4 or 5 foot distant at least, and when ​held to the fire, burned Gloves and Garments at the distance of about three foot from the Fire.
Which were the particulars, the R. Society observed in these Glasses, and gave order to be Registred in their Books; encouraging the Inventor to proceed in this Work with all possible care and diligence, for enabling himself to instruct others in the way of Grinding these Glasses with facility.
The Inventor having declared his resolution to do so, added these Particulars. First, That the Lord Bishop of Salisbury, Seth Ward (who was then absent from the Meeting of the Society) had been by, when the deeper of his two Concaves turned a piece of Wood into flame in the space of ten seconds of time; and the shallower, in five seconds at most, in the season of Autumn, about 9 of the Clock in the Morning, the Weather gloomy*.* This the said judicious Prelate at another Meeting of the Royal Society, attested to be true. Secondly, That the deeper Concave, when held to a lucid Body, would cast a Light strong enough to read by at a considerable distance. Thirdly, That exposing the same to a Northern Window, on which the Sun shined not at all, or very little, he had perceived, that it would warm ones hand sensibly, by collecting the warm'd Air in the day-time, which it would not do after Sun-set.
An Account
1. Our Diurnal Tides, from about the latter end of March till the latter end of September, are about a foot higher (perpendicular, which is always to be understood) in the Evening than in the Morning, that is, in every Tide that happens after 12 in the day before 12 at night.
2. On the contrary, the Morning Tides from Michaelmas 'till our Lady-day in March again, are constantly higher by about a foot than those that happen in the Evening. And this proportion holds in both, after the gradual increase of the Tides ​rising from the Neap to the highest Spring; and the like decrease of its height 'till Neap again is deducted.
3. The highest Menstrual Spring Tide is always the third Tide after the New or Full-Moon, if a cross Wind do not keep the Water out, as the North-east or North-west usually doth; whose contrary Winds, if strong, commonly make those to be High-Tides upon our Southern Coasts, which otherwise would be but low.
4. The highest Springs make the lowest Ebbs: (though I am inform'd by an expert Waterman, that it sometimes happens, that there may be a very low Ebb, though no high Spring, which they term an Out-let, or Gurges of the Sea; as when a great Storm chances off at Sea, and not on the Land.)
5. The Water neither flows nor ebbs: alike, in respect of equal degrees; but its Velocity increaseth with the Tide 'till just at Mid-water, that is, half flown, or at half Flood, at which time the Velocity is strongest, and so decreaseth proportionably 'till High-Water or Full-Sea. As may be guess'd at by the following Scheme, collected from my loose Papers, containing the Observations, as they were made at several times and places; which I rather set down as a standing proportion of degrees in the general, than to adequate every single Flux or Reflux so exactly as to half inches, or the like; but yet it may bear the odd minutes above six hours well enough. And it is farther to be noted, that although this be restained to Plymouth Haven, or the like, where the Water usually riseth about 16 Foot (I say usually, because it may vary in this Port from the lowest Neap to the highest Annual Spring above 7 or 8 Foot) yet it may indifferently serve for other paces, where it may rise as many fathom, or not so high, by a perpetual Addition or Subtraction.
The Scheme it self.
| | | ho. | | foot. | | inch. | | | | ho. | | foot. | | inch. | |
| | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 1 | ——— | 1 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ | | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 1 | ——— | 1 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 2 | ——— | 2 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ | | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 2 | ——— | 2 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| Of Flowing | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 3 | ——— | 4 | ... | 0 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ | Of Ebbing | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 3 | ——— | 4 | ... | 0 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| Of Flowing | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 4 | ——— | 4 | ... | 0 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ | Of Ebbing | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 4 | ——— | 4 | ... | 0 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 5 | ——— | 2 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ | | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 5 | ——— | 2 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
| | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 6 | ——— | 1 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ | | $\scriptstyle {\left\{{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right.}$ | 6 | ——— | 1 | ... | 6 | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \\\ \\\ \ \end{matrix}}\right\}\,$ |
6. The usual number of Tides, or times of High-water from New-Moon to New-Moon, or from Full-Moon to Full-Moon, is 59.
So far the Remarks hitherto mode by this inqusitive person upon the Subject of Tides, who not only promiseth his own continuance for farther Observations, but also his care of recommending the said Tide-Quæries to the constant observation if an intelligent person living just on the Sea-side.
Enquiries and Directions
For the Ant-Iles, or Caribbe-Islands.
In Numb. 23. some Quæries were publish'd for some parts of the West-Indies, and those for other parts refer'd to another opportunity; which presenting it self at this time, we shall here set down such Enquiries for the Ant-Iles, as were collected out of the Relations of several Authors writing of those Islands, such as are the Natural History of the Ant-Iles, written by a French-man; the History of the Barbadoes by Lygon, &c. to the end, that these Queries being considered by such curious persons as frequent those places, and delight in making careful Observations, they may from thence return such Answers, as may either confirm or rectifie the Relations concerning them already extant. The Enquiries are these:
I. Of Vegetables.
1. Whether the juice of the Fruit of the Tree Junipa, being as clear as any Rock-water, yields a brown Violet-dye, and being put twice upon the same place, maketh it ​look black? And whether this Tincture cannot be got out with any Soap, yet disappears of it self in 9 or 10 days? And whether certain Animals, and particularly Hogs and Parrots, eating of this Fruit, have their Flesh and Fat altogether tinged of a Violet colour?
2. Whether Ring-doves, that feed upon the bitter Fruit of the Acomas Tree, have their Flesh bitter also?
3. Whether the Wood of the Acajau Tree, being red, light, and well scented, never rots in Water, nor breeds any Worms, when cut in due season? And whether the Chests and Trunks made thereof, keep Clothes, placed therein, from being Worm-eaten?
4. Whether the Leaves of a certain Tree, peculiarly called Indian-Wood, give such a haut-goust to Meat and Sauces, as if it were a composition of several sorts of Spices?
5. Whether there be such two sorts of the Wood, call'd Savonier, or Soap-wood, of the one of which the Fruit, of the other the Root serveth for Soap?
6. Whether the bark of the Paretuvier-wood tanns as well as Oak-bark?
7. Whether the Root of the Tree Laitus, being brayed and cast into Rivers, maketh Fishes drunk?
8. Whether the Root of the Manioc is so fertile, that one Acre planted therewith, yields so plentiful a crop, as shall feed more people than six Acres of the best Wheat?
9. What Symptoms do usually follow upon taking of the juice of Manioc, or upon the eating of the Juice with the Root, and what effects are thereby produced upon the Body, that infer it to be accounted a rank Poison? Whether any worse Effects, than may be caused by meer Crudity, as by Turnips or Carrots eaten raw, and much more by raw Flesh, in those that are not used thereto; or at most, some such nauseous or noxious quality, as might be corrected in the taking and the preparation; which correction, if effected, might perhaps render the Bread, made of this Manioc, much heartier, the juice being likely to carry off the Spirit and strength, leaving the remainder spiritless?
​10. The Palmetto-Royal being said by Ligon to be a very tall and streight Tree, and so tough, that none of them have been seen blown down, and withall hollow; inall which respect they may serve for special uses, and particular for long Optic Tubes; 'tis much desired, that the largest and longest pieces of them that can be stow'd in a Ship, may be sent over.
11. Whether the Oyl expressed out of the Plant Ricinus or Palma Christi, be used by the Indians to keep them from Vermin? To send over some of that Oyl.
12. Whether in the passage of the Isthmus from Nombre de Dios to Panama, there is a whole Wood full of Sensitive Trees, of which, as soon as they are touch'd, the Leaves and Branches move with a ratling noise, and wind themselves together into a roundish Figure?
13. Whether there be certain Kernels of a Fruit like a white Pear-plum, which are very Purgative and Emetick, but having the thin film which parteth them into halves taken out, they have no such Operation at all, and are as sweet as a ]ordan-Almond?
14. To send over some of the Roots of the Herb, call'd by our French Author L'herbe aux flesches (the Dart-herb,) which being stamped, is said to have the vertue of curing the wounds made with poison'd Darts.
15. To send some of the Grain of the Herb Musk, putting it up carefully in a Box; which being done, it will keep its Musk scent.
16. To send over a Specimen of all Medicinal Herbs, together with their respective Vertues, as they are reputed there: Item particularly, the Prickle-with at the Barbbadoes; Macao, Mastictree, Locust, Black-wood, yellow within; Five-sprig, Tidle-wood, White-wood, Barbadoes-Cedar.
17. Whether the fruit Mancenille of the Mancenillier-Tree, though admirably fair and fragrant, yet is fatal to the Eater, and falling into the Water, kills the Fishes that eat thereof, except Crabs, who yet are said to be dangerous to eat when they have fed upon this fruit? Whether under the Bark of this Tree is contained a certain glutinous Liquor as white as Milk, very dangerous, so that if you chance to rubit, and this juice ​spurt upon the Shirt like a burning; if upon the naked flesh, it will cause a swelling; if into the eye, blindness for several days? And whether the shadow of this Tree be so noxious, that the bodies of Men reposing under it, will swell strangely? And whether the Meat it self, that is boil'd with the fire of this Wood, contracts a malignity, burning the Mouth and Throat? Further, whether the Natives use the milky juice of this Tree, and the Dew falling from it, and the juice of its Fruit, in the composition of the Poison they infect their Arrows with?
II. Of Animals and Insects.
18. Whether the skin of the Tatou, and the little bone in his Tail, do indeed, as is related, cure deafishness, and pains of the Ears? And whether this Animal be proof not only against the Teeth of Dogs, but also against Bullets?
19. Whether the Birds called Canides, be so docile, that some of them learn to speak not only Indian, but also Dutch and Spanish, singing also the Ayres in the Indian Tongue as well as an Indian himself?
And whether the Bird Calibry have a scent as sweet as the finest Amber and Musk? Both which is affirmed by our French Author.
20. To procure some of the fat of the Birds, called Fregati, reputed to be very Anti-paralytical and Anti-podagrical.
21. To send over a Land-pike, which is said to be like the Water-pike, but that instead of Fins it hath four feet, on which it crawls.
22. Whether the skin of the Sea-wolf, which they otherwise call the Requiem, be so ruff and stiff, that they make Files of them, fit to file Wood? And whether it be usually guided by another Fish, that is beautified with such a variety of curious and lively Colours, that one would say, such Fishes were girt with Necklaces of Pearls, Corals, Emerauds, &c.
​23. Whether the skin of Sea-Calfs, otherwise call'd Lamantins, be so hard, when dry'd, that they serve the Indians for Shields?
24. Whether the Ashes of the Fresh-water Tortoises do hinder the falling of the Hair, being powder'd therewith?
25. Whether the Land-Crabs of these Islands do at certain times hide themselves all under ground for the space of 6 weeks, and during that time change and renew their shells? And whether in hiding themselves thus, they do so carefully cover themselves all about with Earth, that the opening thereof cannot at all be perceived, thereby shutting out the Air, by which they might else be annoyed when they are quite naked, after they have shed their shells, there then remaining no other cover on them, but a very thin and tender skin, which by little thickeneth and hardeneth into a Crust, like the old?
26. Whether the Serpents in those parts, that have black and white spots on their backs, be not venomous? To send over some of such Serpents skins.
27. To send over some of the skins of those huge Lizards, they call Ouayamaca, which, when come to their full bigness, are laid to be five foot long, Tail and all: And especially to send some of those that are said to have the scales of their skins so bright and curious, that at a distance they resemble Cloth of Gold and Silver.
28. Whether the shining Flies, called Cucuyes, hide almost all their light, when taken, but when at liberty, afford it plentifully?
29. Whether there be a sort of Bees brown and blew, who make a black Wax, but the Honey in it whiter and sweeter than that of Europe.
30. Whether in those parts the Indians do cure the bitings of Serpents by eating fresh Citron Pills, and by applying the Unguent, made of the bruised Head of the wounding Serpent, and put hot upon the wound?
31. Whether the Wood lice in those Countries generated out of rotten Wood, are able, not only to eat through Trunks in a day or two, and to spoil Linnen, Clothes, and Books, (of which last they are said to spare only what is written or ​printed;) but also to gnaw the props which support the Cottages, that they fall? And whether the remedy against the latter mischief is, to turn the ends of the Wood that is fixed in the ground; or to rub the Wood with the Oyl of that kind of Palma Christi (a Plant) wherewith the Natives rub their Heads to secure them from Vermin.
32. Whether that sort of Vermin called Ravets, spare nothing of what they meet with (either of Paper, Cloths, Linnen, and Woollen) but Silk and Cotton?
33. Whether the little Cirons called Chiques, bred out of dust, when they pierce once into the Feet, and under the Nails of the Toes, do get ground of the whole body, unless they be drawn out betimes? And whether at first they cause but a little itch, but afterwards having pierced the skin, raise a great inflammation in the part affected, and become in a small time as big as Pease, producing innumerable Nits, that breed others.
As to Inquiries, concerning Earths and Minerals, that my be taken out of Numb. 19. and as for such, which concern the constitution of the Air, Winds, and Weather, they are to he met with in Numb. 11.
To which latter sort may be added touching Hurricans, Whether those terrible Winds, which are said to have formerly happen'd in those parts but once in 7 years, do now rage once in two years, and sometimes twice, yea thrice in one year? And whether they are observed never to fall out but about the Autumnal Equinox; as 'tis affirmed, that in the East-Indies beyond the Line, they never happen but about the Vernal? Whether they are preceded with an extream Calm, and the Rain which falls a little before be bitterish and salt? And whether Birds come timely down by whole flocks from the Hills, and hide themselves in the Valleys, lying close to the ground, to secure themselves from the Tempest approaching?
An Account of two Books.
This Book was lately by two excellent persons of the Florentine Virtuosi, viz. Lorenzo Magalotti, and Paulo Falconieri, presented to the Royal Society, in the name of His Highness Prince Leopold of Tuscany, that great Patron to real Philosophy. The Book contains these particulars:
1. An explication of the Instruments, employed in these Experiments.
2. Exp. belonging to the natural pressure of the Air.
3. Exp. concerning artificial Conglaciations.
4. Exp. about natural Ice.
5. Exp. about the change of the capacity of Metal and Glass.
6. Exp. touching the compression of Water.
7. Exp. to prove that there is no positive Lightness.
8. Exp. about the Magnet.
9. Exp. about Amber, and other substances of a vertue Electrical.
10. Exp. about some changes of Colours in divers Fluids.
11. Exp. touching the motions of Sound.
12. Exp. concerning Projectils.
13. Various Experiments.
As all these Heads are very considerable, and of main importance to Philosophy, so doubtless will the handling of them be by competent judges found worthy of these famous Academians del Cimento.
This Tract perused by some very able and judicious Mathematicians, and particularly by the Lord Viscount Brounker, and the Reverend Dr. John Wallis, receiveth the Character of being very ingeniously and very Mathematically written, and well worthy the study of Men addicted to ​that Science: that in it the Author hath delivered a new Method Analytical for giving the Aggregate of an Infinite or Indefinite converging Series: and that from that ground he teaches a Method of Squaring the Circle, Ellipsis, and Hyperbole, by an Infinite Series, thence calculating the true dimensions as near as you please: And lastly, that by the same method from the Hyperbola he calculateth both the Logarithms of any Natural Number assign'd, and vice versa, the Natural Number of any Logarithm given.
Only a few of these Books were printed by the Author for his own use, and that of his Friends, and a Copy sent over whereby to reprint it here, which is now a doing.
The Mathematical Mr. John Collins, upon a more particular examination of this Book, communicated what follows concerning the same.
The Author's Computation of the Area of a Circle agrees with the Numbers of Van Ceulen; and his computation of the supplemental spaces between the Hyperbola and its Asymptote by Parallels to the other Asymptote, is correspondent to what Gregory of St. Vincent and his Commentators Francis Aynscomb and Alphonse de Sarasa have demonstrated concerning the Logorithms, as represented by those spaces, viz. That if one Asymptote be divided into a rank of continual Proportionals, and if parallels to the other Asymptote be drawn passing through the said rank, and be terminated at the Hyperbola, the spaces contain'd between each such pair of Parallels, are equal to each other, and so added or conceived to be one continued space, may represent the Logarithms; or the said Proportionals, fitted in parallel to the divided Asymptote, do the like, by reason that a Rectangle apply'd to the several Terms of a Geometrical Progression increasing, renders another in the same Ratio decreasing. And both performed by the above-mentioned Analytical method of conveying complicated Polygons circumscrib'd and inscrib'd in the sector of a Circle, Ellipsis, or Hyperbola, which he asserts to be quantities like Surds, not absolutely to be express'd in Numbers.
And it being manifest, that the making of the Table of Logarithms is in effect the same thing as the computing of Area's of those supplemental spaces, the Author accordingly applies it ​thereto, and finds the Logarithms of all Primitive Numbers under 1000 by one Multiplication, two Divisions, and the Extraction of the Square Root, but for Prime Numbers greater, much more easily.
Concerning the construction of Logarithms Mr. Nicholas Mercater hath a Treatise, intituled Logarithmotechnia, likewise at the Press, from which the Reader may receive further satisfaction. And as for Primitive Numbers, and whether any odd number proposed less than 100000 be such, the Reader will meet with a satisfactory Table at the end of a Book of Algebra, written in High Dutch by John Henry Rohn, now translated and enriched, and near ready for publick view.
The Area of an Hyperbola not being yet given by any Man, we think fit a little to explain the Author's meaning.
In Figure 1. Let the Curve DIL represent an Hyperbola, whose Asymptotes AO, AK, make the Right Angle OAK, the Author propounds to find the Hyperbolick space ILNK, contained by the Hyperbolical Line IL, the Asymptote KM, and the two Right Lines IK, LM, which are parallel to the other Asymptote AO.
| He puts the Lines IK | = | 1 000 000 000 000 |
| LM | = | 1 000 000 000 000 | 0 |
| AM | = | 1 000 000 000 000 |
| Hence KM | = | 9 000 000 000 000 |
Whence he finds the space LIKM
| to be $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | 230 258 509 299 404 562 401 78681 | too little. |
| to be $\scriptstyle {\left\{{\begin{matrix}\ \\\ \end{matrix}}\right.}$ | 230 258 509 299 404 562 401 78704 | too great. |
Note: If IK be put for an Unit, then LM may represent 10, and HG 1000, and FE 1024: And by what is demonstrated by Gregory of St. Vincent, it holds,
As the space IBLMKI, Is to the Logarithm of LM, to wit, of 10: So is the space IBEFKI, To the Logarithm of the Number represented by the Line EF, to wit, of 1024
The Author by the same method finds the Area of the space GEFH to be 237 165 266 173 160 421 183 067, and the space LIKM abovesaid being taken for the Logarithm of 10, and tripled, is the Logarithm of 1000, the which added to the space now found, makes the sum 69314718055994529141719170, and 1024, being the 10th Power of 2, the 10th part of this number is the Hyperbolical Logarithm of the Numb. 2, to wit, 6931471805599452914171917. And it holds by proportion,
As 23025850929940456249178700, the Logarithm of 10, To 6931471805599452914171917, the correspondent Logarithm of 2: So 1 000 000 000 000 000 000 000 000 0, the Logarithm of 10 in the ​Tables, To 3010299956639811952405804, the Logarithm of 2 in the Tables.
By this means the Area of one Hyperbola being computed, the Area's of all others may be thence argued, as is shewed by Greg. St. Vincent, and Van Schooten in Tractatu de Organica Conicarum sectionum descriptione.
If the Logarithm of 1 be put 0; and of 10, 1,0000000: If between 1 and 10 you conceive 9999999 mean Proportionals interjected; the first is 1,00000023025853.
If the Logarithm of 1 be put 0; and of 10, 100000: If you conceive 99999 mean Proportionals between 1 and 10, the first is 1,0000235853; if an infinite rank of these be continued, there is no number proposed, but will go nigh to be found in this rank, and the number of Terms, by which it is removed from Unit, is the Logarithm of the Number so found. The Ratio of 1, to 1,00000023025853, some call Elementum Logarithmicum. See Cavallieri's Trigonometry.
The Area of an Hyperbola is frequently required in Gauging; as admit it were required to compute the Solidity of the Segment of an upright Cone cut by a Plain, that would cut the produced opposite Cone; in any such Case the Section is an Hyperbola. But we will only take the Instance, when it is parallel to the Axis.
In Figure II.1. Let BVA represent such a Cone, VC its Avis, BSAR the Circle in the Base. And first, suppose this Cone cut by a Plain passing through the Vertex and the Base USRU; then is the whole Cone divided into such proportions as the Area of the Circle in the Base. Whence we discover the use and the want of a good Table of Area’s of Segments; the best of which kind yet extant is in Sibrand Hantz his Century of Geometrical Problems, translated out of Dutch into English by Captain Thomas Rudd, who omitted the said Table; useful likewise for finding the Area of the Segment of an Ellipsis, and the obtaining the quantity of Liquor out of, or left in a Cask part empty.
And we hint, that a Table of Natural Versed Sines is to be found in Maginus, and of Logarithmical ones in Cavallieri's Directorium Universale Uranometricum.
2. The former Plain did cut out a Chord-line in the Base, to wit, SR; through the same imagine another Plain to pass, and to cut the Cone beneath the Vertex, as at O:, then is the Wedge contain'd between both these Plains (to wit, VSROV) equal to 13 of that Cylindrick or Prismatick Figure, whose Altitude is equal to the Perpendicular VP falling from the Vertex of the Cone to the cutting Plain, and whose Base SORTS is the Area of the Figure cut; in this case, an Hyperbola: When the Plain passeth parallel to the side BV, a Parabola; when it will meet with VB produced, a Portion of an Ellipsis. By this means, if a Brewer's Tun (taken to be a Circular Truncus Coni) lean, and be not cover'd over with Liquor in its bottom, it may be computed by subtracting the two known before​mention'd parts out of the whole: If it stand upright, and be divided by an upright Plain into two Partitions, imagine it to be a whole Cone, and first, by the method above, find the Segment, as of the whole, and afterwards of the additional Top-Cone, the difference of those two gives the Content of the correspondent Partition.
3. But if the Liquor cut both sides, the Tun leaning, as BCDE, in Figure III. Suppose BAE to be the Triangle through the Axis of the whole Cone, then the Elliptick Cone ACD to the whole ABE is in a Triplicate Ratio of the Side-line AB or AE, to the Geometrical Mean between AC and AD, that is,
As the Cube of the Side-line AB, Is to the Solidity of the whole Cone ABE: So is the Cube of the Geometrical Mean between AC and AD, To the Solidity of the Elliptick Cone ACD.
And this readily follows from the Doctrine of Viviani de Maximis & Minimis, where 'tis demonstrated, that any such Elliptick Cones, cut out of an Upright Cone, that have the Area's of their Triangles through the Axis equal, are equal to each other; and likewise to that Upright Cone which hath the same Area on its Triangle through the Axis on the former Plain thereof; and these Area's he calls their right Canons.
And the mean Proportional by 23. E. 6. finds the sides of an Isosceles Triangle in the Plain of the Axis equal to the Scalene Triangle; and then these Cones are to each other in a Triplicate Ratio of their Axes, Side-lines, or Base-lines, which are proportional to their Axes.
The Area of an Hyperbola being obtain'd, the Solidity of the Hyperbolical Fusa or Spindles (made by the rotation of an Hyperbola about its Base) and their Trunci are computed, according to Cavallieri (in his Geometrical Exercises, printed at Bononia 1647.) and the solid Zones of these Figures may be well taken to represent a Cask.
In the SAVOY,