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Pure mathematics consider magnitude generally, simply, and abstractedly; whence they have exclusively the prerogative of being founded on the elementary notions of quantity. This class comprehends 1, Arithmetic, or the art of computation; 2, Geometry, which teaches us to measure extension; 3, Analysis, or the calculation of magnitudes in general; and 4, mixed Geometry, or the combination of common geometry with analysis.
Mixed mathematics borrow from physics one or more incontestable experiments, or suppose in bodies some principal and necessary quality; and then, by a methodical and demonstrative chain of reasoning, they deduce, from the principle established, conclusions as evident and certain, as those which pure mathematics draw immediately from axioms and definitions. To this class belong 1, Mechanics, or the science of the equilibrium and motion of solid bodies; 2, Hydrodynamics, in which the equilibrium and motion of fluids are considered; 3, Astronomy, or the science of the motions of the celestial bodies 4, Optics, or the theory of the effects of light; 5, and lastly, Acoustics, or the theory of sound.
I have here arranged the different parts of mathematics in that order which appears to me best calculated for exhibiting at one view their reciprocal concatenation, in the state in which they are at present; but this order is not altogether analogous to their actual and historical developement.
It is not possible to fix the origin of mathematics with precision; though we are able to affirm, that it goes back to the remotest ages. When mankind, relinquishing
linquishing the wandering and savage life, united in societies; and general laws or conventions had established it as a rule, that each should provide for his own subsistence, without seizing what was in the possession of another; want and convenience, the two great springs of industry, soon invented the most necessary arts. Huts were built; iron was forged; the land was divided; the course of the stars was observed. It was seen, that the earth yielded spontaneously, and in every season, various productions for the food of fome animals; but that for others, of still greater utility, and in greater abundance, it required the assistance of a cultivation regulated by the seasons: thus the ground was sown, and the harvest reaped.
All these observations, all these performances, at first extremely rude and unskilful, were connected with mathematics by a secret and unknown tie: though for a long time they had no rule The or guide but experience and blind custom. assiduous labour required in hunting, fishing, and the business of the field, did not allow men to ascend to general and abstruse ideas: the circle of their physical wants bounded that of their thoughts. By imperceptible degrees, several of them acquiring a sort of superfluity, either by superiour industry or abundant harvests, gave themselves up to that idle ness, to which all animals have a natural propensity. Happiness they imagined was to be found in this state of indolence and repose; a seductive illusion, which soon undeceives us, but to which we are at least indebted for the first flights of the human understanding in those days. The languor of inaction, the
torment of wearisomeness annexed to it, and the ac tivity of the thinking principle which we carry within us, snatched man from this disgraceful lethargy, and gave an impulse to that spirit of curiosity and research, which incessantly agitates us, and which demands sustenance no less imperiously than the body itself.
Man then beheld with new eyes the magnificent spectacle, which Nature exhibited on all sides to his senses and imagination: he learned to examine things, and compare them together. Ideas acquired from physical objects were detached from them as it were, and transported to an intellectual world: orators, poets, painters arose: the phenomena of nature were studied with discriminating attention, and the mind was impressed with a desire to know the causes, by which they were produced. Geometry, confined at first to the art of measuring the fields, was extended to new purposes, and proposed to itself loftier and more difficult problems. Astronomy was enriched by regular observations, and by several instruments adapted to increase their number, and to give them the requisite accuracy and connexion. Machines were invented, in which a skilful combination of wheels and levers was employed to. raise or transport the heaviest loads: in a word, all parts of the mathematics successively advanced. Their progress would have been more rapid, if fanaticism and the insatiate love of power, while they ravaged the Earth, had not too frequently obscured the flame of genius for a long series of ages: but, as a fire concealed beneath the embers, it resumed it's lustre in happier times, and by degrees the edifice of
science arose. Let us hope, that posterity will feel the honourable ambition of pursuing the work, without being discouraged by the apprehension of never being able to complete the roof.
The most general and best established opinion is, that mathematics began to acquire a certain solidity among the primitive chaldeans, and the primitive egyptians, that is to say the two most ancient people known, almost at the same period. According to a permanent tradition, transmitted from age to age, the shepherds of Chaldea, dwelling under a clear sky, laid the foundations of astronomy, during the leisure of their quiet occupation. If their observations were too imperfect, to serve as the basis of any theory, they at least afforded some general hints to the first astronomers, and saved them the trouble of some mistaken attempts.
The magi, or priests of Egypt, directed by the laws of their institution, to study and collect the secrets of nature, were become the depositaries and dispensers of all human knowledge, From all parts men came to consult them, and to acquire from them instruction. They would have deserved the unbounded respect and gratitude of the world, if, content to enlighten it, they had not sometimes endeavoured to impose upon it likewise, and to conceal the proud desire of sway under the sacred veil of religion.
Nations, like individuals, scek to swell their origin, and carry it backward to remote ages. The chinese and hindoos are particularly accused of this national mania. If we believe their own accounts, they were
the first inventors of all the sciences, and of all the arts. As they rest their pretensions more particularly on the antiquity of astronomy among them, I shall defer my examination of them, till I come to speak more at large of the progress of this science.
With the mathematics of the ancients we are acquainted only through the writings of the greeks: and to estimate the instructions, which these derived from their intercourse with the magi, we possess not the necessary documents. Some authors have said, that Thales, in one of his visits to Memphis, taught the egyptians how to measure the height of the pyramids by the extent of their shadow, a propósition that ranks very low in the elements of geometry. If this were true, we must infer, that the egyptians were but little versed in the science: but the fact is not probable, and it is the safest way to assert nothing on the subject, since all the records of egyp‐ tian science perished with the alexandrian library. We ought simply to admit, that, if the egyptians were the first masters of the greeks, they were soon surpassed by their scholars. As soon as the mathematics began to take root in Greece, we see them shoot up with a strong and rapid growth, and successively enrich themselves with a number of important discoveries, in which the mutual connexion, of principles and consequences marks the unity and continuance of the plan. The greeks became in some measure the preceptors of all other nations: they alone have had the glory of excelling in every branch, in the military art, poetry, oratory, painting, the accurate sciences, &c. Most of the illus