# Reflections upon Ancient and Modern Learning/Chapter 14

CHAP. XIV.
Of Ancient and Modern Geometry and Arithmetick.
In the Method which I set to my self in these Reflections, I chose to begin with an Enquiry into those Sciences, whose Extent is more liable to be contested; and so onwards, to those which may more easily be determined. Monsieur Perrault, who has not finished his Parallel, that I know of, took it for granted, that if the Prize were granted to the Moderns in Eloquence, in Poesie, in Architecture, in Painting, and in Statuary, the Cause would be given up in every Thing else; and he, as the declared Advocate for the Moderns, might go on triumphantly with all the rest. Wherein, possibly, he was not, in the main, much mistaken. How he manages the remaining Part of his Parallel, I know not. I intend to begin with Abstracted Mathematicks; both because all its Propositions are of Eternal Truth, and besides, are the Genuine Foundations upon which all real Physiology must be built.
​The Method which I shall follow is this: (1.) I shall enquire into the State of Ancient and Modern Mathematicks, without any particular Application of the Properties of the several Lines and Numbers, Surfaces and Solids, to Physical Things. (2.) I shall enquire what new Instruments have been invented, or old ones improved, by which the Knowledge of Nature of any sort has been, or may be, further enlarged. (3.) I shall enquire whether any Improvements have been actually made of Natural History, and of any Physico-Mathematical or Physical Sciences, such as Astronomy, Musick, Opticks, Medicks, and the like. (4.) From all this, I shall endeavour to pass a Judgment upon the Ancient and Modern Ways of Philosophizing concerning Nature in general, and its principal Phænomena, or Appearances.
I begin with Geometry and Arithmetick, because they are general Instruments whereby we come to the Knowledge of many of the abstrusest Things in Nature; since, as Plato said of old, God always Geometrizes in all his Works. That this Comparison might be the more exact, I desired my learned and worthy Friend, Mr. John Craige, to give me his Thoughts upon this Matter: His own learned Wri​tings upon the most difficult Parts of Geometry, for such are the Quadratures of Curve Lines, will be sufficient Vouchers for his Skill in these Things. I shall set down what he says, in his own Words.
'If we take a short View of the Geometry of the Ancients, it appears, that they considered no Lines, except Streight Lines, the Circle, and the Conick Sections: As for the Spiral, the Quadratrix, the Conchoid, the Cissoid, and a few others, they made little or no Account of them. It is true, they have given us many excellent and useful Theorems concerning the Properties of these others; but far short of what has been discovered since. Thus the Quadrature of the Circle, which did so much exercise and perplex the Thoughts of the Ancients; How imperfect is that of Archimedes, in comparison of that exhibited by Van Ceulen? And every Body knows how this is exceeded by the later Performances of Mr. Newton, and Monsieur Leibnitz. Archimedes, with a great deal of Labour, has given us the exact Quadrature of the Parabola; but the Rectification of the Parabolick Line, depending on the Quadrature of the Hyperbola, is the Invention of this last Age. The rare Properties of the Conick Se​ctions, in the Reflexion and Refraction of Light, are the undoubted Discoveries of these later Times. It were easie to give more Instances of this Nature, but these are sufficient to shew how far the Modern Mathematicians have out-done the Ancients, in discovering the noblest and usefullest Theorems, even of those few Figures which they chiefly considered.'
'But all this is nothing, in Comparison of that boundless Extent which the Modern Mathematicians have carried Geometry on to: Which consists in their receiving into it all the Curve Lines in Nature, together with the Area's and Solids that result from them; by distinguishing them into certain Kinds, and Orders; by giving general Methods of describing them, of determining their Tangents, their Lengths, their Area's, and the Solids made by the Rotation of them about their Axes. Add to all this, the general Methods that have been invented of late for finding the Properties of a great Number of these Curves, for the Advancement of Opticks, Mechanicks, and other Parts of Philosophy: And let any Man of Sense give the Preference to the Ancient Geometry if he can.'
​'That the Ancients had general Methods of Constructing all plain Problems by a streight Line and a Circle, as also all Solid Problems by the help of a Conick Section, is most certain. But it is as certain that here they stopped, and could go no further, because they would not receive any Order of Curves beyond the Conick Sections, upon some nice Scrupulosity in multiplying the Number of the Postulata, requisite to the describing of them. Whereas the Modern Geometers, particularly the renowned Des Cartes, have given general Rules for Constructing all Problems of the 5th. or 6th. Degree. Which Method, if rightly understood, is applicable to all Problems of any Superior Order.'
'How deficient the Geometry of the Ancients was in that Part which related to the Loca Geometrica, is manifest from the Account that Pappus gives us of that Question, about which Euclid and Apollonius made so many ineffectual Attempts: The Solution whereof we owe entirely to Mr. Isaac Newton (i).(i) Philos. p. 74, 75. For it is evident that Des Cartes mistook the true Intent of the Ancients in this Matter. So that the Loca Solida is now one of the perfectest Parts of Geometry that we have; which before ​was one of the most confused, and defective.'
'From comparing the Ancient and Modern Geometry, I proceed to the Comparison of those Arts, to which we owe the Improvements both of the one, and the other. These are chiefly Two, viz. Algebra, and the Method of Indivisibles. As to the latter of these, I shall not stand to enquire whether Cavallerius was the first Inventor, or only the Restorer of it. (k) History of Algebra, pag. 285.I know (k) Dr. Wallis is of Opinion that it is nothing but the Ancients Method of Exhaustions, a little disguised. It is enough for your Purpose, that by the help of Cavallerius's Method, Geometry has been more promoted in this last Age, than it was in all the Ages before. It not only affords us neat and short Demonstrations, but shews us how to find out the abstrusest Theorems in Geometry. So that there has hardly been any considerable Improvement of late, which does not owe its Rise to it; as any Man may see, that considers the Works of Cartes, Fermat, Van Heuraet, Huygens, Neil, Wallis, Barrow, Mercator, Leibnitz, and Newton. Archimedes's Propositions of the Properties of a Sphere and a Cylinder, are some of the easiest Examples of this Method. How vastly ​more curious, and more useful Theorems have been since added to Geometry, is known to every one that is conversant in the afore-mentioned Authors; especially Mr. Newton, Leibnitz and Huygens: To instance particulars, were to transcribe their whole Books and Treatises.'
'Let us, in the next Place, compare the Ancient and Modern Algebra. That the Ancients had some kind of Algebra, like unto ours, is the Opinion of several learned Writers of late: And it is evident from the Seven remaining Books of Diophantus, that it was brought to a considerable Length in his Time. But how infinitely short this was of that Algebra which we now have, since Vieta's Time, will appear to any that considers the different Process of both. For, tho' Diophantus has given us the Solution of a great many hard and knotty Arithmetical Problems, yet the last Step of his Resolution serves only for one particular Example of each Problem: So that for every new Example of the same Question, there must be a new Process made of the whole Analysis. Whereas by our Modern Algebra, the Analysis of any one Case gives a general Canon for all the infinite Cases of each Problem; ​whereby we discover many curious Theorems about the Properties of Numbers, not to be attained by Diophantus's Method; this being the peculiar Advantage of Specious Algebra, first introduced by Vieta, and wonderfully promoted by several worthy Mathematicians since. Beside this intolerable Imperfection of the Ancient Algebra, used by Diophantus, which required as many different Operations as the Problem had different Examples, that is, infinite; all which are included in one general Solution by the Modern Algebra; there is this great Defect in it, that in Undetermined Questions, which are capable of innumerable Solutions, Diophantus's Algebra can seldom find any more than one; whereas, by the Modern Algebra, we can find innumerable, sometimes all in one Analysis; though in many Problems we are obliged to re-iterate the Operation for every new Answer. This is sufficient to let you see, that (even in the Literal Sense) our Algebra does infinitely exceed that of the Ancients. Nor does the Excellency of our Algebra appear less in the great Improvements of Geometry. The reducing all Problems to Analytical Terms, has given Rise to those many excellent ​Methods, whereby we have advanced Geometry infinitely beyond the Limits assigned to it by the Ancients. To this we owe, (1.) The Expressing all Curves by Equations, whereby we have a View of their Order, proceeding gradually on in infinitum. (2.) The Method of Constructing all Problems of any Assignable Dimension; whereas the Ancients never exceeded the Third. Nay, from the Account which Pappus gives us of the afore-mentioned Question, it is evident, that the Ancients could go no further than Cubick Equations: For he says expresly, they knew not what to make of the continual Multiplication of any Number of Lines more than Three; they had no Notion of it. (3.) The Method of Measuring the Area's of many Infinities of Curvilinear Spaces; whereas Archimedes laboured with great Difficulty, and wrote a particular Treatise of the Quadrature of only one (l)(l) The Parabola., which is the simplest and easiest in Nature. (4.) The Method of Determining the Tangents of all Geometrick Curve Lines; whereas the Ancients went no further than in determining the Tangents of the Circle and Conick Sections. (5.) The Method of Determining the Lengths of an infinite Num​ber of Curves; whereas the Ancients could never measure the Length of one. If I should descend to Particulars, the Time would fail me. As our Algebra, so also our Common Arithmetick is prodigiously more perfect than theirs; of which, Decimal Arithmetick and Logarithms are so evident a Proof, that I need say no more about it.'
'I would not be thought, however, to have any Design to sully the Reputation of those Great Men, Conon, Archimedes, Euclid, Apollonius, &c. who, if they had lived to enjoy our Assistance, as we now do some of theirs, would, questionless, have been the greatest Ornaments of this Age, as they were deservedly the greatest Glory of their own.' Thus far Mr. Craig.
Those that have the Curiosity to see some of these Things proved at large, which Mr. Craig has contracted into one View, may be amply satisfied in Dr. Wallis's History of Algebra, joyned with Gerhard Vossius's Discourses De Scientiis Mathematicis.
It must not here be forgotten, that Abstracted Mathematical Sciences were exceedingly valued by the ancientest Philosophers: None that I know of expressing a Contempt of them but Epicurus, tho' ​all did not study them alike. Plato is said to have written over the Door of his Academy, Let no Man enter here, who does not understand Geometry. None of all the learned Ancients has been more extolled by other learned Ancients, than Archimedes. So that if in these Things the Moderns have made so great a Progress, this affords a convincing Argument, that it was not Want of Genius which obliged them to stop at, or to come behind the Ancients in any Thing else.
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