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fired twenty-two years in Egypt) was attributed found out by him; the most certain way of intenthe invention of the thirty-second and forty-seventh tion and reasoning. Upon the door of his acapropositions of the first of Euclid; for the latter of demy was read this inscription, dis dywuésprü Be which he conceived so much joy, that he is said toitw. Thirteen of his familiar acquaintance are to have offered an hecatomb. A discovery of commemorated by Proclus, as inen by whose stuthis kind, in later times, would have been entitled dies the mathematics were improved. After these but to a small share of honour, and the want of were Leon, and Eudoxus of Cnidos, a man great knowing these propositions must needs make their in arithmetic, and to whom we owe the whole fifta geometry very coarse and imperfect. Upon this book of the elements; Xenocrates, and Aristotle. account, therefore, it may be concluded that the To Aristeus, Isidore, and Hypsicles, most subtile learning of the Egyptians, for which their priests geometricians, we are indebted for the books of were so famous, and Moses so celebrated in holy solids. Afterwards Euclid gathered together the writ for having attained it, did not so much cone inventions of others, dispused them into order, sist in mathematics, as in the arts of legislation, improved them, and denionstrated them more acand civil polity, and magic. Their magicians, or curately, and left to us those Elements, by which wise men, thought that the sun, moon, stars, and youth is every where instructed in the mathemaelements, were appointed to govern the world; and tics. He died two hundred and eighty-four years though they acknon ledged that God might, upon before Christ. Almost an bundred years after extraordinary occasions, work miracles, reveal his followed Eratosthenes and Archimedes; the write will by audible voices, divine appearances, dreams ings of the first are lost, but we have many remains or prophecies, yet they thought also, that, general- of the latter. The very name ofArchimedes suggests ly speaking, oracles were given, prodigies caused, an idea of the top of human subtilty, and the perdreams of things to come occasioned by the dis- fection of the whole mathematical sciences; his position of the several parts of the universe to wonderful inventions have been delivered to us by influence upon one another, at the proper places Polybius, Plutarch, Tzetres, and others. He was and seasons, as constantly and as necessarily as the the first who was able to give the exact quadrature heavenly bodies performed their revolutions; and or mensuration of a space, bounded by the arch they imagined that their learned professors, by a of a curve and a right line, which he did by dedeep study of, and profound inquiry into the monstrating that the segment of a parabola is to powers of nature, could make themselves able to its inscribed triangle as 4:3. Cotemporary with work wonders, obtain oracles and omens, and in him was Conon; and at no great distance of time terpret dreams, either from fate (meaning the was Apollonius of Perga, another prince in geonatural course of things), or from nature, wbich metry, called, by way of encomium, the great geowas when they used any artificial assistance by metrician. We have extant four books of conics drinks, inebriations, discipline, or other means, in his name; though some think Archimedes was which were thought to have a natural power to the author of them: we have also three books of produce the vaticinal influence, or prophetic frenzy: spherics by Theodosius the Tripolite. In the year and in all these particulars they thought the seventy, after Christ, appeared Claudius Ptoles Deity not concerned, but that they were the mere mæus, the prince of astronomers, a inan not only natural effects of the influence of the elements most skilful in astronomy, but in geometry also, and planets at set times and critical junctures. as many other things by him written witness,
From Egypt, geometry travelled into Greece; but especially his books of subtenses. After these for Thales the Milesian, who flourished five hun: flourished Eutocius, Ctesibius, Proclus, Pappus, dred and eighty-four years before Christ, was the and Theon. Then ensued a long period of ignofirst of the Greeks who, coming into Egypt, trans- rance; arts and sciences, liberty and learning ferred geometry from thence into Greece. He is being driven away and overrun by that brutish reputed, certainly, besides other things, to have herd of northern barbarians, whose whole excelfound out the fifth, fifteenth, and twenty-sixth lenee was in their bones and muscles, and feats of propositions of Euclid's first book, and the second, chivalry their bighest ambition. During this disma) third, fourth, and fifth of the fourth book. The night of ignorance, doubtless many curious discosume person improved astronomy, for be began toveries and useful pieces of knowledge were totally observe the equinoxes and solstices, and was the lost, and the remainder buried, as it were, in ruins, first who foretold an eclipse of the sun.
till the restoration of learning upon the taking After him was Pythagoras, of Samos, before- of Constantinople by the Turks in the year 1451 mentioned. This man much improved and adorn- after Christ; whereby the residue of Greek and ed the mathenatic sciences, and so attached was Roman learning was driven for refuge into Italy he to arithmetic in particular, that almost his and the other neighbouring countries of Europe. whole method of philosophizing was taken from Geometry has always been valued for its extennumbers. He first of all abstracted geometry sive usefulness, but has been most adınired for its from matter, in which elevation of mind he found true and real excellence, which consists in its perout several of Euclid's propositions. He first laid spicuity and perfect evidence. It may, therefore, open the matter of incommensurable magnitudes, be of use to consider the nature of the demonand the five regular bodies.
strations, and the steps by which the ancients Next fourished Anaxagoras of Clazomenæ, and were able, in several instances, from the mensuraEnopides of Chios. These were followed by tion of right-lined figures, to judge of such as were Briso, Antipho, and Hippocrates, of Chios; which bounded by curve lines; for as they did not allow three, for attempting the quadrature of the circle, themselves to resolve curvilinear figures into rectivere reprehended by Aristotle, and, at the same linear elements, it is worth while to examine by time, celebrated. Then came Democritus, Thco- what art they could make a transition from the dorus, Cyrenæus, and Plato, than whoin no one ove to the other. brought greater lustre to the mathematical sciences; They found that similar triangles are to each he amplified geometry with great and potable ad- other in the duplicate ratio of their bomologous ditions, bestowing incredible study upon and sides; and by resolving similar polygons into siabove all, the art analytie, or of resolution, was milar triangles, the same proposition was extended to these polygons also. But when they came quantities must be the same as the invariable sato compare curvilineal figures, which cannot be tio of the two variable quantities: and this may resolved into rectilineal parts, this method failed. be considered as the most simple and fundamentál Circles are the only curvilineal plain figures cou- proposition in this doctrine, by which we are ensidered in the elements of geometry. If they abled to compare curvilineal spaces in some of the could have allowed themselves to bave considered more simple cases. these as similar polygons of an infinite number of The next improvement in the way of demonsides, (as some have since done, who pretend to strating among the ancient geometricians, seems abridge their demonstrations,) after proving that to be that which we call the method of exany similar polygons inscribed in circles are in the haustions. See EXHAUSTIONS. duplicate ratio of their diameters, they would have Archimedes, indeed, takes a rather different way immediately extended this to the circles them- for comparing the spheroid with the cone and cyselves, and would have considered 2 Euc. 12. as an linder, that is more general, and has a nearer anaeasy corollary from the first: but there is reason logy to the modern methods. He supposes the to think they would not have admitred a demon- terms of a progression to increase constantly by stration of this kind, for the old writers were very the same difference, and demonstrates several careful to admit no precarious principles, or aught properties of such a progression relating to the else but a few self-evident truths, and no demon- sum of the terms, and the sum of their squares; strations but such as were accurately deduced from by which he is able to compare the parabolic cothem. It was a fundamental principle with them, noid, the spheroid, and hyperbolic conoid, with that the difference of any two unequal quantities, the cone; and the area of his spiral line with the by which the greater exceeds the Tesser, may be area of the circle. There is an analogy betwixt added to itself till it shall exceed any proposed what he has shewn of these progressions, and the finite quantity of the same kind : and that they proportions of figures demonstrated in the elefounded their propositions concerning curvilineal mentary geometry; the consequence of which may figures upon this principle, in a particular manner, illustrate his doctrine, and serve, perhaps, to sbew is evident from the demonstrations, and from the that it is more regular and complete in its kind express declaration of Archimedes, who acknow- than some have imagined. The relation of the ledges it to be a foundation upon which he esta- sum of the terms to the quantity that arises by blished his own di-courses, and cites it as assumed taking the greatest of them as often as there are by the ancients in demonstrating all the propo. terms, is illustrated by comparing the triangle sitions of this kind: but this principle seems to be with a parallelogram of the same height and base; inconsistent with the admitting of an infinitely and what he bas demonstrated of the sum of the little quantity or difference, which, added to itself squares of the terms compared with the square of any number of times, is never supposed to become the greatest term may be illustrated by the proequal to any finite quantity soever,
portion of the pyramid to the prism, or of the cone They proceeded, therefore, in another manner, to the cylinder, their bases and heights being less direct indeed, but perfectly evident. They equal; and by the ratios of certain frustums or found that the inscribed similar polygons, by proportions of these solids deduced from the ele. having the number of their sides increased, conti- mentary proportions. nually approached to the areas of the circles; so He appears solicitous, that his demonstrations that the decreasing difference between each circle should be found to depend on those principles and its inscribed polygon, by still further and fur- only that had been universally received before his ther divisions of the circular arches, which the time. In his treatise of the quadrature of the sides of the polygon subtend, could become less parabola, he mentions a progression, whose terms than any quantity could be assigned; and that all decrease constantly in the proportion of four to this while the similar polygons observed the same one; but he does not suppose this progression to constant invariable proportions to each other, viz. be continued to infinity or mention the sum of an that of the squares of the diameters of the circles. infinite number of terms; though it is manifest, Upon this they founded a demonstration, that the that all which can be understood by those who as. proportion of the circles themselves could be no sign that sum was fully known to him. He apother than that same invariable ratio of the simi. pears to have been more fond of preserving to the lar inscribed polygons. For they proved, by the science all its accuracy and evidence, than of doctrine of proportions only, that the ratio of the advancing paradoxes; and contents himself with two inscribed polygons cannot be the same as the demonstrating this plain property of such a proratio of one of the circles to a magnitude less gression, that the sum of the terms continued at than the other, nor the same as the ratio of one of pleasure, added to the third part of tbe last term, the circles to a magnitude greater than the other; amounts always to of the first term: nor does he therefore the ratio of the circles to each other, suppose the chords of the curve to be bisected to must be the same as the invariable ratio of the si- infinity; so that after an infinite bisection, the inmilar polygons inscribed in them, which is the cribed polygon might be said to coincide with duplicate of the ratio of the diameters.
the parabola. These suppositions would have In the same manner the ancients have demon- been new to the geometricians in his time, and strated, that pyramids of the same height are to such he appears to have carefully avoided. each other as their bases, that spheres are as the This is a summary account of the progress that cubes of their diameters, and that a cone is the was made by the ancients in measuring and comone third part of a cylinder on the same base, and paring curvilineal figures, and of the method by of the same height.' In general, it appears from which they demonstrated their theorems of this their way of demonstration, that when two varia- kind. It is often said, that curve lines have been ble quantities, which always have an invariable considered by them as polygons of an infinite ratio to each other, approach at the same time to number of sides; but this principle no where aptwo determined quantities, so that they may dif- pears in their writings: we never find them re. fer less from them than by any assignable mea. solving any figure or solid into infinitely small sare; the ratio of these limits or determined elements; on the contrary, they seem to bare
aroided such suppositions, as if they judged them Notwithstanding, as this method is here ex. unfit to be received into geometry, when it was plained, it is manifestly founded on inconsistent obvious, that their demonstrations might have and impossible suppositions ; for while the lines, been sometimes abridged by admitting them. of which surfaces are supposed to be made up, They considered curvilineal areas as the limits of are real lines of no breadth, it is obvious, that no circumscribed or inscribed figures, of a inore sim- number wbatever of them can furm the least ple kind, which approach to these limits (by a bi- imaginable surface: if they are supposed to be of section of lines or angles, continued at pleasure) some sensible breadth, in order to be capable of so that the diff-rence between them may become filling up spaces, i. e. in reality to be paralleloless than any given quantity. The inscribed or grams, bow minute soever be their altitude, the circumscribed figures were always conceived to surfaces may not be to each other in the proporbe of a maguitude and number that is assignable;
tion of all such lines in one to all the like lines in and from what had been shewn of these águres, the other; for surfaces are not always in the same they demonstrated the mensuration, or the pro- proportion to each other with the parallelograurs portions of the curvilineal limits themselves, by inscribing them. arguments ab absurdo. They had made frequent The same contradictory suppositions obvio use of demonstrations of this kind from the begin- ously attend the composition of sol.ds by parallel aing of the elements; and these are, in a particu- planes, or of lines by such imaginary points. lar manner, adapted for making a transition This beterogeneous composition of quantity, from right-lined figures, to such as are bounded and confusion of its species, so different from that by curve lines. By admitting them only, they distinctness, for which the mathematics were ever established the more difficult and sublime part of famous, was opposed at its first appearance by their geometry, on the same foundation as the several eminent geometricians : particularly by first elements of the science; nor could they have Guldinus and Tacquet ; who not only excepted to proposed to themselves a more perfect inodel. the first principles of this method, but taxed the
But as these demonstrations, by determining conclusions formed upon it as erroneous. But as distinctly all the several magnitudes and propor. Cavalerius took care that the threads or lines of tions of these inscribed and circumscribed figures, which the surfaces to be compared together were did frequently extend to very great lengths, other formed, should have the same breadth in each (as methods of demonstrating have been contrived by he hiinself expresses it) the conclusions deduced the moderns, whereby to avoid these circumstan- by his method might generally be verified by tial deductions. The first attempt of this kind sounder geometry; since the comparison of these known to us, was made by Lucas Valerius; but lines was, iu effect, the comparing together the afterwards Cavalerius, an lialian, about the year inscribed figures. one thousand six hundred and thirty-tive, advance
As in the application of this method, error, by ed his method of Indivisibles, in which he pro- proper caution, might be avoided, the assistance poses, not only to abbreviate the ancient demon- it seemed to promise in the analytical part of strations, but to remove the indirect form of geometry made it eagerly followed by those who reasoning used by them, of proving the equality were more desirous to discover new propositions, or proportion between lines and spaces, from the than solicitous about the elegance or propriety of ito possibility of their having any different reia- their demonstrations. Yet so strange did the con. tion; and to apply to these curved magnitudes tradictory conception appear, of composing surthe same direct kind of proof that was before ap- faces out of lines, and solids out of planes, that, plied to right-lined quantities.
in a short time, it was new modelled into that This method of comparing magnitudes, invent- form, which it still retains, and which now uni. ed by Cavalerius, supposes lines to be compounded versally prevails among the foreign mathemati of points, surfaces of lines, and solids of planes; cians, under the name of the differential method, er, to make use of his own description, surfaces or the analysis of infinitely littles. are considered as cloth, consisting of parallel
In this reformed notion of indivisibles, surfaces threads; and solids are considered as formed of are now supposed as composed not of lines bút parallel planes, as a book is composed of its leaves, of parallelograms, having infinitely little breadths; with this restriction, that the threads or lines, of and solids, in like manner as toimed of prisis, which surfaces are compounded, are not to be of having infinitely little altitudes By this al eraany conceivable breadth, nor the leaves or planes tion it was imagined, that the heterogeneous conof solids of any thickness. He then forms these position of Cavalerius was 'sufficiently evaded, propositioos, that surfaces are to each other, as and all the advantages of his method retained. all the lines in one to all the lines in the other; Bat here, again, the same absurdity occurs as beand solids, in like manuer, in the proportion of fore; for if, by the infinitely little breadth of these all the planes.
parallelograms, we are to understand wbat these This method exceedingly shortened the former words literally import, i. e. no breadth at all; then tedious demonstrations, and was easily perceived; they cannot, any more than the lines of Cavaleso that problems, which at first sight appeared of rius, compose a surface; and if they have any an insuperable difficulty, were afterwards resolved, breadth, the right lines bounding them cannot and came, at length, to be despised, as too simple coincide with a surface bounded by a curve line. aod easy: the mensuration of parabolas, hyper- The followers of this new method grew bolder bolas, spirals of all the higher orders, and the fa- than the disciples of Cavalerius, and having transmous cycloid, were among the early productions formed his points, lines, and planes, into infiniteof this period. The discoveries made by Torri- ly little lines, surfaces, and solids, they pretended celli, de Fermat, de Roberva!, Gregory Śt. Vio- they no longer compared together heterogeneous cent, &c. are well known. They who have not quantities, and insisted on their principles, being read many authors, may find a synopsis of this now become genuine; but the mistakes they fremethod in Ward's Young Mathematician's Guide, qnently fell into were a sufficient confutation of where he treats of the mensuration of superficies their boasts; for notwithstanding this new model, and solids.
the same limitations and cautions were
cessary : for instance, this agreement between the treatise (as he tells 'as), before he had examined inscribing figures and the curved spaces to which the writings of Archimedes ; and he proposes his they are adapted, is only partial; and in applying theorems and demonstrations in a less accurate their principles to propositions already determin- form : he supposes the progressions to be contied by a juster method of reasoning, they easily nued to infinity, and investigates, by a kind of in. perceived this defect; both in surfaces and solids, duction, the proportion of the sum of the powers, it was evident, at first view, that the perimeters to the production that would arise by taking the disagreed. And as no oue instance can be given, greatest power as often as there are terms. His where these indivisible or infinitely little parts do demonstrations, and some of his expressions, (as so completely coincide with the quantities they when he speaks of quantities more than infinite) are supposed to compound, as in every circum- have been excepted against ; however, it must be stance to be taken for them, without producing owned, this valuable treatise contributed to proerroneous conclusions, so we find, where a surer duce the great improvements which soon after guide was wanting, or disregarded, these figures followed. were often imagined to agree, where they ought to The next promoter of geometry, with respect have been supposed to differ.
to time, among our countrymen, was Dr. BarLeibnitz, in two dissertations, one on the re- row, a man of a penetrating genius, and very in. sistance of fluids, the other on the motion of the defatigable: he had amassed a large magazine heavenly bodies, has, on this principle, reasoned of learving; and his general character was, that falsely conceruing the lines intercepted between whatever subject he treated he exhausted : be curves and their tangents. Bernoulli bas, like was a perfect master of the ancient geometry; and wise, made the saine mistake in a dissertation on has obliged us with compendious, yet clear de. the resistance of fluids, and in a pretended solu- monstrations of what is left of the geometrical tion of the problem concerning isoperimetrical writings of Euclid, Archimedes, Apollonius, and curves. Nay, Mr. Parent has bad the rashness Theodosius. But the advances he made in curreto oppose erroneous deductions from this absurd lined geometry, his own particular improvements, principle, to the most indubitable demonstrations are contained in his lectures. He begins with of the great Huygens. Thus it appears, that the treating on the generation of magnitude, which doctrine of indivisibles contains an erroneous me- comprehends the original of mathematical hypo-, thod of reasoning, and, in consequence thereof, theses. Magnitude may be produced various in every new subject to which it shall be applied, ways, or conceived so to be; but the primary and is liable to fresh errors.
chief among them is that performed by local mo. It is also manifest, that the great brevity it gave tion, which all of them must in some sort suppose; to demonstrations, arose entirely from the absurd because, without motion, nothing can be generated attempt of comparing curvilineal spaces in the or produced : so true is Aristotie's axiom, viz. be same direct manner as right line figures can be that is ignorant of motion, is necessarily ignorant compared; for, in order to conclude directly the of nature. What mathematicians chiefly consider equality or proportion of such spaces, no scruple in motion, are these two properties, viz. the mode was made of supposing, contrary to truth, that of lation or manner of bearing; and the quantity rectilineal figures, capable of such direct compa- of the motive force. From these springs the difrison, could adequately fill up the spaces in ques. ferences of motions flow; but because the quantity tion; whereas, the doctrine of exhaustions does of motive force cannot be known without time, the not attempt, from the equality or proportion of doctor gives a long metaphysical account of the the inscribing or circumscribing figures, to con- nature of time; which he defines to be, abstract. clude, directly, the like proportions of these edly, the capacity or possibility of the continuance of any spaces, because those figures can never, in reality, thing in its own being. Towards the latter end be made equal to the spaces they are adapted to: of this he agrees with Aristotle, that we not only but as these figures may be made to differ from measure motion by time, but also time by motion; the spaces to which they are adapted, by less than because they determine each other : for in like any space proposed, how minute soever, it shews manner, as we first of all measure a space by some by a just, though indirect deduction from these magnitude, and declare it is so much; and aftercircunscribing and inscribing figures, that the wards, by means of this space, compute other mag. spaces whose equality is to be proved, can have nitudes correspondent with it: so we first assume no difference; and that the spaces, whose propor- time from some motion, and afterwards judge tion is to be shewn, cannot have a different pro- thence of other motions, which, in reality, is no portion thau that assigned them.
more than comparing some motions with others, The Arithmetica Infinitorum of Dr. Wallis was by the assistance of time; just as we investigate the fullest treatise of this kind that appeared be the ratios of magnitude by the help of some space. fore the invention of Quxions. Archimedes had E. g. He who computes the proportion of motion considered the sums of the terms in an arithmeti. by the proportion of time, does no more than get cal progression, and of their squares only, (or ra- the said ratio of notions from clocks, dials, or ther the limits of these sums only) these being suf- from the proportion of solar motions in the same ficient for the mensuration of the figures he had time. Again, because time is a quantity uniformly exainined. Dr. Wallis treats this subject in a extended, all whose parts correspond, either provery general inamier, and assigns like limits for portionally to the respective parts of an equal the sums of any powers of the terms, whether the motion, or to the parts of spaces moved through exponents be integers or fractions, positive or ne- with an unequal motion; it may, therefore, be gative. Having discovered one general theorem very aptly represented to our ininds by any magthat includes all of this kind, he then composed nitude alike in all its parts; and especially the ner progressions from various aggiegates of these most simple ones, such as a straight or circular terms, and enquired into the suins of the powers line; between which and time there happens to of these terms, by which he was enabled to mea- be inuch likeness and analogy: for as time consure accurately, or by approximation, the areas of sists of parts altogether similar, it is reasovable to figures without number: but he composed this consider it as a quantity endowed witli one dimen.
sion only; whether we imagine it to be made up, and without mixture; therefore, if right lines pas as it were, either of the simple addition of rising rallel to each other, and horizontal, be drawp moments, or of the continual Aux of one moment; through all the points of the perpendicular line reand for that reason ascribe only length to it, and presenting the time, the plain superficies thence determine its quantity by the length of a line resulting, will exactly represent the aggregate of passed over. As a line is looked on to be the trace the degrees of velocity; which superficies, baving of a point moving forward, being in some sort divi- its parts proportionable to the respective parts of sible by a point, and may be divided by mo- the space moved through, may very well represent tion one way, viz. as to length; so time may be that space. If the velocities answering to each conceived as the trace of a moment continually instant of time are equal, this superficies will be a flowing; having some kind of divisibility from an parallelogram; if unequal, a triangle. From the instant, and from a successive flux, inasmuch as properties of the former figure are deduced all the it can be divided some way or other. And like as theorems of equable and uniform motion; and the quantity of a line consists of but one length from the latter, all those which concern equally following a motion, so the quantity of time parsues accelerated motion. Moreover, if the degrees of but one succession stretched out, as it were, in velocity, in a continual succession from rest, length; which the length of the space moved over throughout every instant of time, to a given des shews and determines. Time may, therefore, al- gree, be conceived to increase to it, or decrease ways be expressed by a right line; first, indeed, from thence to rest in the progression of the taken or laid down at pleasure; but whose parts square numbers, the aggregatical velocity, as well will exactly answer to the proportionable parts of as the space moved through, may most conveniente time, as its points do the respective instants of ly be represented by the semiparabola, whose vertex time, and will aptly serve to represent them. denotes rest, the several equal parts of the absciss,
The doctor next proceeds to the effective force the given equal times, and the ordinates, the reof time, being the same as what he before called spective degrees of velocity; from a well known the motive force by which magnitudes are gene, property of the parabola. In like manner, any rated. He considers this as a kind of quantity, supposed degrees of velocity, any way increasing capable of computation, like other quantities : for or decreasing continually, or interruptedly after it is plain from experience, that when two move any imaginable way, may be truly and conveable bodies depart from the same place along the niently expressed by right lines applied to that resame line, the one moves a greater space than the presenting the time, keeping whatever proportion other in the same time; the reason of which can any one is pleased to assign; so that kuowing only be this, that that body which moves swiftest, from thence the representative space,
the quantity is acted upon by a greater force or motive power; of space moved through will be easily had, and this force, therefore, admitting of greater and the contrary; for it is easy to deduce theorems, if less modifications, may be justly conceived as any one knows rightly and congruously how to divisible into any infinite or indefinite parts; the reduce quantities of any kind soever, subject to least of which is called rest, or the lowest degree of his contemplation, to analogous magnitudes. velocity: considering, therefore, the thing abso- Perhaps tbis dry account of these metaphysical lutely, in order to represent the quantity of this subjects may to some scem tedious; but if it be force justly to the mind, we need only lay down considered that hereupon are founded the theories sound regular magnitude in its stead. As a right the descent of heavy bodies, of pendulums, and line is the most simple and perspicuous of any, it of projectiles, the reducing of which to geometriis, therefore, the fittest to represent any degree cal deinonstrations raised the famous Galileo to so thereof. When this force comes under a mathe, high a reputation, that he was said to have added matical consideration, it is called velocity; which two new sciences to the mathematics; and when is defined to be that power by which a moveable body we further consider that the doctrine of fusions is can pass over a given space in a given time; whence it comprised in two mechanical problems, and that follows, that every particular quantity of any ve- mechanics, or the doctrine of motion, depends locity cannot be known, neither by the space upon computation of the quantities of time, velomored through only, nor by the time singly, but cities, and forces ; it will then appear that these may be found by calculation from the quantity of considerations directly lead towards fluxions, and space and tiine together; as, on the contrary, the that it cannot be time ill-spent to consider their quantity of time may be obtained from the quan- nature abstractedly; unless we could be content to tity of the space and velocity together: nor does know the manner of operating by them only. the quantity of space (so far as it can be known without contemplating the reason of them in this way by motion) depend wholly upon the theory, and knowing whether they are really quantity of a definite velocity, or upon any as- scientific or not. See FLUXIONS, INDIVISIBLES, signed time, but upon the conjoint ratio of both. &c. The quantity of space is found after the same For a catalogue of the principal writers, ancient manner as we do that of a superficies, by its die and modern, we must refer the reader to the armensions; but the quantities of velocity and time ticle Geometry in Dr. Hutton's Mathematical are found exactly after the same inanner as when Dictionary. a superficies and one of its dimensions are given, Geometry is distinguished into theoretical or me thereby find the other: for to every moment of speculative, and practical. time there answers some degree of velocity, which Theoretical or Specululite Geometry, treats of the a moveable body is then conceived to have; i) which various properties and relations in magnitudes, degree some length of the space mored over au- demonstrating the theorems, &c. And swers. When time flows equally, it wil be most Practical Geometry, is that is hich applies those aptly represented by a right line; and the several speculations and theorems to particular vses in degrees of velocity, whether equal or unequal, ineach the solution of probleins, and in the measureinstaut, may be also expressed by right lines; and ments in the ordinary concerns of life. because these degrees of velocity do in every ino- Speculative geometry again may be divided into weat of time pass over one another independently elementary and sublime.