But a curve of the first gender (because a right line cannot be reckoned among the curves) is the same with a line of the second order; and a curve of the second gender, the same with a line of the third order; and a line of an infinitesimal order is that, which a right line may cut in infinite points; as the spiral, helicoid, the quadratrix, and every line generated by the infinite revolutions of a radius. It is to be observed that it is not so much the equation, as the construction or description, that makes any curve, geometrical, or not. Thus, the circle is a geometrical line, not because it may be expressed by an equation, but because its description is a postulate: and it is not the simplicity of the equation, but the easiness of the description, that is to determine the choice of the lines for the construction of a problem. The equation that expresses a para bola, is more simple than that which expresses a circle; and yet the circle, by reason of its more simple construction, is admitted before it. Again, the circle and the conic sections, with respect to the dimensions of the equations, are of the same order; and yet the circle is not numbered with them in the construction of problems, but by reason of its simple description is depressed to a lower order, viz. that of a right line; so that it is not improper to express that by a circle, which may be expressed by a right line; but it is a fault to construct that by the conic sections which may be constructed by a circle. GEOMETRICAL PROGRESSION, a progression in which the terms have all successively the same ratio: as 1, 2, 4, 8, 16, &c. where the common ratio is 2. The general and common property of a geometrical progression is, that the product of any two terms, or the square of any one single term, is equal to the product of every other two terms that are taken at an equal distance on both sides from the former. So of these terms, 1, 2, 4, 8, 16, 32, 64, &c. 1 X 64 2×32 = 4 × 16=8×8=64. In any geometrical progression, if a denote the least term, cube. 9 GEOMETRICAL PROPORTION, called also simply proportion, is the similitude or equality of ratios. Thus, if ab::c:d, or abc:d, the terms a, b, c, d, are in geometrical proportion; also 6, 3, 14, 7, are in geometrical proportion, because 6:3:14:7, or 6:3=14:7. In a geometrical proportion, the product of the extremes, or 1st and 4th terms, is equal to the product of the means, or 2d and 3d terms. So ad = bc, and 6 × 7 = 3 × 14 = 49. GEOMETRICALLY. ad. (from geometrical.) According to the laws of geometry (Ray). GEOMETRICIAN. s. (ews.) One skilled in geometry; a geometer (Brown). To GEOMETRIZĖ. v. n. (ywil.) To act according to the laws of geometry (Boyle). GEOMETRY, the science or doctrine of extension, or extended things; that is of lines, surfaces, or solids. The word is Greek, yupia, formed of you, or earth; and perfew, measure; it being the neces sity of measuring the earth, and the parts and places thereof, that gave the first occasion to the invention of the principles and rules of this art; which has since been extended and applied to numerous other things; insomuch that geometry with arithmetic form now the general foundation of all mathematics. Who were the first inventors of geometry is by no means certain. It is generally allowed that the Chaldeans were first possessed of the mathe matical sciences, especially astronomy, which must imply geometry. Whether Abraban taught these sciences first to the Egyptians, when he went from Ur of the Chaldees, as some learned men assert, is not clear; but on this we may depend, that the Egyptians were the first people that cultivated geometry, being compelled thereto by necessity, the mother of inventions, in order to ascertain to every man his legal property and estate, in a country where boundaries and land-marks were swept away and confounded by yearly inundations. That the Egyptians, in their ancient, free, monarchical state were acquainted with some of the simple elements and easy problems in geometry, is not denied; but we cannot believe they made any great improvements in the abstruse parts thereof, since to Pythagoras (the famous philosopher of Samos, who flourished so low as about five hundred and twenty years before Christ, and who had lived twenty-two years in Egypt) was attributed the invention of the thirty-second and forty-seventh propositions of the first of Euclid; for the latter of which he conceived so much joy, that he is said to have offered an hecatomb. A discovery of this kind, in later times, would have been entitled but to a small share of honour, and the want of knowing these propositions must needs make their geometry very coarse and imperfect. Upon this account, therefore, it may be concluded that the learning of the Egyptians, for which their priests were so famous, and Moses so celebrated in holy writ for having attained it, did not so much consist in mathematics, as in the arts of legislation, and civil polity, and magic. Their magicians, or wise men, thought that the sun, moon, stars, and elements, were appointed to govern the world; and though they acknowledge that God might, upon extraordinary occasions, work miracles, reveal his will by audible voices, divine appearances, dreams or prophecies, yet they thought also, that, generally speaking, oracles were given, prodigies caused, dreams of things to come occasioned by the disposition of the several parts of the universe to influence upon one another, at the proper places and seasons, as constantly and as necessarily as the heavenly bodies performed their revolutions; and they imagined that their learned professors, by a deep study of, and profound inquiry into, the powers of nature, could make themselves able to work wonders, obtain oracles and omens, and interpret dreams, either from fate (meaning the natural course of things), or from nature, which was when they used any artificial assistance by drinks, inebriations, discipline, or other means, which were thought to have a natural power to produce the vaticinal influence, or prophetic frenzy and in all these particulars they thought the Deity not concerned, but that they were the mere natural effects of the influence of the elements and planets at set times and critical junctures. From Egypt, geometry travelled into Greece; for Thales the Milesian, who flourished five hundred and eighty-four years before Christ, was the Arst of the Greeks who, coming into Egypt, transferred geometry from thence into Greece. He is reputed, certainly, besides other things, to have found out the fifth, fifteenth, and twenty-sixth propositions of Euclid's first book, and the second, third, fourth, and fifth of the fourth book. The same person improved astronomy, for he began to observe the equinoxes and solstices, and was the first who foretold an eclipse of the sun. After him was Pythagoras, of Samos, beforementioned. This man much improved and adorned the mathematic sciences, and so attached was he to arithmetic in particular, that almost his whole method of philosophizing was taken from numbers. He first of all abstracted geometry from matter, in which elevation of mind he found out several of Euclid's propositions. He first laid open the matter of incommensurable magnitudes, and the five regular bodies. Next flourished Anaxagoras of Clazomenæ, and Enopides of Chios. These were followed by Briso, Antipho, and Hippocrates, of Chios; which three, for attempting the quadrature of the circle, were reprehended by Aristotle, and, at the same time, celebrated. Then came Democritus, Theodorus, Cyrenæus, and Plato, than whom no one brought greater lustre to the mathematical sciences; he amplified geometry with great and notable additions, bestowing incredible study upon it, and above all, the art analytic, or of resolution, found out by him; the most certain way of invention and reasoning. Upon the door of his academy was read this inscription, dus ảyswμíze̟n1@ air. Thirteen of his familiar acquaintance are commemorated by Proclus, as men by whose studies the mathematics were improved. After these were Leon, and Eudoxus of Cnidos, a man great in arithmetic, and to whom we owe the whole fifth book of the elements; Xenocrates, and Aristotle. To Aristeus, Isidore, and Hypsicles, most subtile geometricians, we are indebted for the books of solids. Afterwards Euclid gathered together the inventions of others, disposed them into order, improved them, and demonstrated them more accurately, and left to us those Elements, by which youth is every where instructed in the mathematics. He died two hundred and eighty-four years before Christ. Almost an hundred years after followed Eratosthenes and Archimedes; the writings of the first are lost, but we have many remains of the latter. The very name of Archimedes suggests an idea of the top of human subtilty, and the perfection of the whole mathematical sciences; his wonderful inventions have been delivered to us by Polybius, Plutarch, Tzetres, and others. He was the first who was able to give the exact quadrature or mensuration of a space, bounded by the arch of a curve and a right line, which he did by demonstrating that the segment of a parabola is to its inscribed triangle as 4:3. Cotemporary with him was Conon; and at no great distance of time was Apollonius of Perga, another prince in geometry, called, by way of encomium, the great geometrician. We have extant four books of conics in his name; though some think Archimedes was the author of them: we have also three books of spherics by Theodosius the Tripolite. In the year seventy, after Christ, appeared Claudius Ptolemæus, the prince of astronomers, a man not only most skilful in astronomy, but in geometry also, as many other things by him written witness, but especially his books of subtenses. After these flourished Eutocius, Ctesibius, Proclus, Pappus, and Theon. Then ensued a long period of ignorance; arts and sciences, liberty and learning, being driven away and overrun by that brutish herd of northern barbarians, whose whole excellence was in their bones and muscles, and feats of chivalry their highest ambition. During this dismal night of ignorance, doubtless many curious discoveries and useful pieces of knowledge were totally lost, and the remainder buried, as it were, in ruins, till the restoration of learning upon the taking of Constantinople by the Turks in the year 1451 after Christ; whereby the residue of Greek and Roman learning was driven for refuge into Italy and the other neighbouring countries of Europe. Geometry has always been valued for its extensive usefulness, but has been most admired for its true and real excellence, which consists in its perspicuity and perfect evidence. It may, therefore, be of use to consider the nature of the demonstrations, and the steps by which the ancients were able, in several instances, from the mensuration of right-lined figures, to judge of such as were bounded by curve lines; for as they did not allow themselves to resolve curvilinear figures into rectilinear elements, it is worth while to examine by what art they could make a transition from the one to the other. They found that similar triangles are to each other in the duplicate ratio of their homologous sides; and by resolving similar polygons into similar triangles, the same proposition was extend ed to these polygons also. But when they came to compare curvilineal figures, which cannot be resolved into rectilineal parts, this method failed. Circles are the only curvilineal plain figures considered in the elements of geometry. If they could have allowed themselves to have considered these as similar polygons of an infinite number of sides, (as some have since done, who pretend to abridge their demonstrations,) after proving that any similar polygons inscribed in circles are in the duplicate ratio of their diameters, they would have immediately extended this to the circles themselves, and would have considered 2 Euc. 12. as an easy corollary from the first: but there is reason to think they would not have admitted a demonstration of this kind, for the old writers were very careful to admit no precarious principles, or aught else but a few self-evident truths, and no demonstrations but such as were accurately deduced from them. It was a fundamental principle with them, that the difference of any two unequal quantities, by which the greater exceeds the lesser, may be added to itself till it shall exceed any proposed finite quantity of the same kind: and that they founded their propositions concerning curvilineal figures upon this principle, in a particular manner, is evident from the demonstrations, and from the express declaration of Archimedes, who acknowledges it to be a foundation upon which he established his own discourses, and cites it as assumed by the ancients in demonstrating all the propositions of this kind: but this principle seems to be inconsistent with the admitting of an infinitely little quantity or difference, which, added to itself any number of times, is never supposed to become equal to any finite quantity soever. They proceeded, therefore, in another manner, less direct indeed, but perfectly evident. They found that the inscribed similar polygons, by having the number of their sides increased, continually approached to the areas of the circles; so that the decreasing difference between each circle and its inscribed polygon, by still further and further divisions of the circular arches, which the sides of the polygon subtend, could become less than any quantity could be assigned ; and that all this while the similar polygons observed the same constant invariable proportions to each other, viz. that of the squares of the diameters of the circles. Upon this they founded a demonstration, that the proportion of the circles themselves could be no other than that same invariable ratio of the simiJar inscribed polygons. For they proved, by the doctrine of proportions only, that the ratio of the two inscribed polygons cannot be the same as the ratio of one of the circles to a magnitude less than the other, nor the same as the ratio of one of the circles to a magnitude greater than the other; therefore the ratio of the circles to each other must be the same as the invariable ratio of the similar polygons inscribed in them, which is the duplicate of the ratio of the diameters. In the same manner the ancients have demonstrated, that pyramids of the same height are to each other as their bases, that spheres are as the cubes of their diameters, and that a cone is the one-third part of a cylinder on the same base, and of the same height. In general, it appears from their way of demonstration, that when two variable quantities, which always have an invariable ratio to each other, approach at the same time to two determined quantities, so that they may diffor less from them than by any assignable measure the ratio of these limits or determined quantities must be the same as the invariable ratio of the two variable quantities: and this may be considered as the most simple and fundamental proposition in this doctrine, by which we are enabled to compare curvilineal spaces in some of the more simple cases. The next improvement in the way of demonstrating among the ancient geometricians, secmis to be that which we call the method of ex. haustions. See EXHAUSTIONS. Archimedes, indeed, takes a rather different way for comparing the spheriod with the cone and cylinder, that is more general, and has a nearer analogy to the modern methods. He supposes the terms of a progression to increase constantly by the same difference, and demonstrates several properties of such a progression relating to the sum of the terms, and the sum of their squares; by which he is able to compare the parabolic conoid, the spheroid, and hyperbolic conoid, with the cone; and the area of his spiral line with the area of the circle. There is an analogy betwixt what he has shewn of these progressions, and the proportions of figures demonstrated in the elementary geometry; the consequence of which may illustrate his doctrine, and serve, perhaps, to shew that it is more regular and complete in its kind than some have imagined. The relation of the sum of the terms to the quantity that arises by taking the greatest of them as often as there are terms, is illustrated by comparing the triangle with a parallelogram of the same height and base; and what he has demonstrated of the sum of the squares of the terms compared with the square of the greatest term may be illustrated by the proportion of the pyramid to the prism, or of the cone to the cylinder, their bases and heights being equal; and by the ratios of certain frustums or proportions of these solids deduced from the ele mentary proportions. He appears solicitous, that his demonstrations should be found to depend on those principles only that had been universally received before his time. In his treatise of the quadrature of the parabola, he mentions a progression, whose terms decrease constantly in the proportion of four to one; but he does not suppose this progression to be continued to infinity, or mention the sum of an infinite number of terms; though it is manifest, that all which can be understood by those who assign that sum was fully known to him. He ap pears to have been more fond of preserving to the science all its accuracy and evidence, than of advancing paradoxes; and contents himself with demonstrating this plain property of such a progression, that the sum of the terms continued at pleasure, added to the third part of the last term, amounts always to of the first term: nor does he suppose the chords of the curve to be bisected to infinity; so that after an infinite bisection, the inscribed polygon might be said to coincide with the parabola. These suppositions would have been new to the geometricians in his time, and such he appears to have carefully avoided. This is a summary account of the progress that was made by the ancients in measuring and com paring curvilineal figures, and of the method by which they demonstrated their theorems of this kind. It is often said, that curve lines have been considered by them as polygons of an infinite number of sides; but this principle nowhere ap pears in their writings: we never find them resolving any figure or solid into infinitely small elements on the contrary, they seem to have avoided such suppositions, as if they judged them unft to be received into geometry, when it was obvious, that their demonstrations might have been sometimes abridged by admitting them. They considered curvilineal areas as the limits of circumscribed or inscribed figures, of a more simple kind, which approach to these limits (by a bisection of lines or angles, continued at pleasure) so that the difference between them may become less than any given quantity. The inscribed or circumscribed figures were always conceived to be of a magnitude and number that is assignable and from what had been shewn of these figures, they demonstrated the mensuration, or the proportions of the curvilineal limits themselves, by arguments ab absurdo. They had made frequent use of demonstrations of this kind from the beginning of the elements; and these are, in a particular manner, adapted for making a transition from right-lined figures, to such as are bounded by curve lines. By admitting them only, they established the more difficult and sublime part of their geometry, on the same foundation as the first elements of the science; nor could they have proposed to themselves a more perfect model. Bat as these demonstrations, by determining distinctly all the several magnitudes and proportions of these inscribed and circumscribed figures, did frequently extend to very great lengths, other methods of demonstrating have been contrived by the moderns, whereby to avoid these circumstantial deductions. The first attempt of this kind known to us, was made by Lucas Valerius: but afterwards Cavalerius, an Italian, about the year one thousand six hundred and thirty-five, advanced his method of Indivisibles, in which he proposes, not only to abbreviate the ancient demonstrations, but to remove the indirect form of reasoning used by them, of proving the equality or proportion between lines and spaces, from the impossibility of their having any different relation; and to apply to these curved magnitudes the same direct kind of proof that was before applied to right-lined quantities. This method of comparing magnitudes, invented by Cavalerius, supposes lines to be compounded of points, surfaces of lines, and solids of planes; or, to make use of his own description, surfaces are considered as cloth, consisting of parallel threads; and solids are considered as formed of parallel planes, as a book is composed of its leaves, with this restriction, that the threads or lines, of which surfaces are compounded, are not to be of any conceivable breadth, nor the leaves or planes of solids of any thickness. He then forms these propositions, that surfaces are to each other, as all the lines in one to all the lines in the other; and solids, in like manner, in the proportion of all the planes. This method exceedingly shortened the former tedious demonstrations, and was easily perceived; to that problems, which at first sight appeared of an insuperable difficulty, were afterwards resolved, and came, at length, to be despised, as too simple and easy; the mensuration of parabolas, hyperbolas, spirals of all the higher orders, and the famous cycloid, were among the early productions of this period. The discoveries made by Torricelli, de Fermat, de Roberval, Gregory St. Vincent, &c. are well known. They who have not read many authors may find a synopsis of this method in Ward's Young Mathematician's Guide, where he treats of the mensuration of superficies and solida. Notwithstanding, as this method is here explain ed, it is manifestly founded on inconsistent and impossible suppositions; for while the lines, of which surfaces are supposed to be made up, are real lines of no breadth, it is obvious, that no number whatever of them can form the least imaginable surface: if they are supposed to be of some sensible breadth, in order to be capable of filling up spaces, i. . in reality to be parallelograms, how minute soever be their altitude, the surfaces may not be to each other in the proportion of all such liges in one to all the like lines in the other; for surfaces are not always in the same proportion to each other with the parallelograms inscribing them. The same contradictory suppositions obviously attend the composition of solids by parallel planes, or of lines by such imaginary points. This heterogeneous composition of quantity, and confusion of its species, so different from that distinctness, for which the mathematics were ever famous, was opposed at its first appearance by several eminent geometricians: particularly by Guldinus and Tacquet; who not only excepted to the first principles of this method, but taxed the But as conclusions formed upon it as erroneous. Cavalerius took care that the threads or lines of which the surfaces to be compared together were formed should have the same breadth in each (as he himself expresses it) the conclusions deduced by his method might generally be verified by sounder geometry; since the comparison of these lines was, in effect, the comparing together the inscribed figures. As in the application of this method, error, by proper caution, might be avoided, the assistance it seemed to promise in the analytical part of geometry made it eagerly followed by those who were more desirous to discover new propositions than solicitous about the elegance or propriety of their demonstrations. Yet so strange did the contradictory conception appear, of composing surfaces out of lines, and solids out of planes, that, in a short time, it was new modelled into that form, which it still retains, and which now universally prevails among the foreign mathematicians, under the name of the differential method, or the analysis of infinitely littles. In this reformed notion of indivisibles, surfaces are now supposed as composed not of lines but of parallelograms, having infinitely little breadths; and solids, in like manner as formed of prisms, having infinitely little altitudes. By this alteration it was imagined, that the heterogeneous composition of Cavalerius was sufficiently evaded, and all the advantages of his method retained. But here, again, the same absurdity occurs as before; for if, by the infinitely little breadth of these parallelograms, we are to understand what these words literally import, i. e. no breadth at all; then they cannot, any more than the lines of Cavalerius, compose a surface; and if they have any breadth, the right lines bounding them cannot coincide with a surface bounded by a curve line. The followers of this new method grew bolder than the disciples of Cavalerius, and having transformed his points, lines, and planes, into infinitely little lines, surfaces, and solids, they pretended they no longer compared together heterogeneous quantities, and insisted on their principles, being now become genuine; but the mistakes they frequently fell into were a sufficient confutation of their boasts; for notwithstanding this new model, the same limitations and cautions were still ne eessary for instance, this agreement between the inscribing figures and the curved spaces to which they are adapted, is only partial; and in applying their principles to propositions already determined by a juster method of reasoning, they easily perceived this defect; both in surfaces and solids, it was evident, at first view, that the perimeters disagreed. And as no one instance can be given, where these indivisible or infinitely little parts do so completely coincide with the quantities they are supposed to compound, as in every circumstance to be taken for them, without producing erroneous conclusions, so we find, where a surer guide was wanting, or disregarded, these figures were often imagined to agree, where they ought to have been supposed to differ. Leibnitz, in two dissertations, one on the resistance of fluids, the other on the motion of the heavenly bodies, has, on this principle, reasoned falsely concerning the lines intercepted between curves and their tangents. Bernoulli has, likewise, made the same mistake in a dissertation on the resistance of fluids, and in a pretended solution of the problem concerning isoperimetrical curves. Nay, Mr. Parent has had the rashness to oppose erroneous deductions from this absurd principle, to the most indubitable demonstrations of the great Huygens, Thus it appears, that the doctrine of indivisibles contains an erroneous method of reasoning, and, in consequence thereof, in every new subject to which it shall be applied, is liable to fresh errors. It is also manifest, that the great brevity it gave to demonstrations arose entirely from the absurd attempt of comparing curvilineal spaces in the same direct manner as right line figures can be compared; for, in order to conclude directly the equality or proportion of such spaces, no scruple was made of supposing, contrary to truth, that rectilineal figures, capable of such direct comparison, could adequately fill up the spaces in question; whereas, the doctrine of exhaustions does not attempt, from the equality or proportion of the inscribing or circumscribing figures, to conclude, directly, the like proportions of these spaces, because those figures can never, in reality, be made equal to the spaces they are adapted to but as these figures may be made to differ from the spaces to which they are adapted, by less than any space proposed, how minute soever, it shews by a just, though indirect deduction from these circumscribing and inscribing figures, that the spaces whose equality is to be proved, can have no difference; and that the spaces, whose proportion is to be shewn, cannot have a different proportion than that assigned them. The Arithmetica Infinitorum of Dr. Wallis was the fullest treatise of this kind that appeared be fore the invention of fluxions. Archimedes had considered the sums of the terms in an arithmetical progression, and of their squares only, (or rather the limits of these sums only) these being sufficient for the mensuration of the figures he had examined. Dr. Wallis treats this subject in a very general manner, and assigns like limits for the sums of any powers of the terms, whether the exponents be integers or fractions, positive or negative. Having discovered one general theorem that includes all of this kind, he then composed new progressions from various aggregates of these terms, and enquired into the sums of the powers of these terms, by which he was enabled to measure accurately, or by approximation, the areas of figures without number: but he composed this treatise (as he tells us) before he had examined the writings of Archimedes; and he proposes his theorems and demonstrations in a less accurate form: he sup poses the progressions to be continued to infinity, and investigates, by a kind of induction, the proportion of the sum of the powers, to the production that would arise by taking the greatest power as often as there are terms. His demonstrations, and some of his expressions, (as when he speaks of quantities more than infinite) have been excepted against; however, it must be owned, this valuable treatise contributed to produce the great improvements which soon after followed. The next promoter of geometry, with respect to time, among our countrymen, was Dr. Barrow, a man of a penetrating genius, and very indefatigable: he had amassed a large magazine of learning; and his general character was, that whatever subject he treated he exhausted: he was a perfect master of the ancient geometry; and has obliged us with compendious, yet clear demonstrations of what is left of the geometrical writings of Euclid, Archimedes, Apollonius, and Theodosius. But the advances he made in curve-lined geometry, his own particular improvements, are contained in his lectures. He begins with treating on the generation of magnitude, which comprehends the original of mathematical hypotheses. Magnitude may be produced various ways, or conceived so to be; but the primary and chief among them is that performed by local mo tion, which all of them must in some sort suppose; because, without motion, nothing can be generated or produced: so true is Aristotle's axiom, viz. he that is ignorant of motion, is necessarily ignorant of nature. What mathematicians chiefly consider in motion, are these two properties, viz. the mode of lation or manner of bearing; and the quantity of the motive force. From these springs the differences of motions flow; but because the quantity of motive force cannot be known without time, the doctor gives a long metaphysical account of the nature of time; which he defines to be, abstractedly, the capacity or possibility of the continumice of any thing in its own being. Towards the latter endi of this he agrees with Aristotle, that we not only measure motion by time, but also time by motion; because they determine each other: for in like manner, as we first of all measure a space by some magnitude, and declare it is. so much; and afterwards, by means of this space, compute other magnitudes correspondent with it: so we first assume time from some motion, and afterwards judge thence of other motions; which, in reality, is no more than comparing some motions with others, by the assistance of time; just as we investigate the ratios of magnitude by the help of some space. E. g. He who computes the proportion of motion by the proportion of time, does no more than get the said ratio of motions from clocks, dials, or from the proportion of solar motions in the same time. Again, because time is a quantity uniformly extended, all whose parts correspond, either proportionally to the respective parts of an equal motion, or to the parts of spaces moved through with an unequal motion; it may, therefore, be very aptly represented to our minds by any magnitude alike in all its parts; and especially the most simple ones, such as a straight or circular line; between which and time there happens to be much likeness and analogy: for as time consists of parts altogether similar, it is reasonable to consider it as a quantity endowed with one dimen |