MATHEMATICS

, the science of quantity; or a science that considers magnitudes either as computable or measurable.

The word in its original, maqhs<*>s, mathesis, signifies discipline or science in general; and, it seems, has been applied to the doctrine of quantity, either by way of eminence, or because, this having the start of all other sciences, the rest took their common name from it.

As to the origin of the Mathematics, Josephus dates it before the flood, and makes the sons of Seth observers of the course and order of the heavenly bodies: he adds, that to perpetuate their discoveries, and secure them from the injuries either of a deluge or a conflagration, they had them engraven on two pillars, the one of stone, the other of brick; the former of which, he says, was yet standing in Syria in his time.

Indeed it is pretty generally agreed that the first cultivators of Mathematics, after the flood, were the Assyrians and Chaldeans; from whom, Josephus adds, the science was carried by Abraham to the Egyptians; who proved such notable proficients, that Aristotle even fixes the first rise of Mathematics among them. From Egypt, 584 years before Christ, Mathematics passed into Greece, being carried thither by Thales; who having learned geometry of the Egyptian priests, taught it in his own country. After Thales, came Pythagoras; who, among other Mathematical arts, paid a particular regard to arithmetic; drawing the greatest part of his philosophy from numbers. He was the first, according to Laertius, who abstracted geometry from matter; and to him we owe the doctrine of incommensurable magnitude, and the five regular bodies, besides the first principles of music and astronomy. To Pythagoras succeeded Anaxagoras, Oenopides, Briso, Antipho, and Hippocrates of Scio; all of whom particularly applied themselves to the quadrature of the circle, the duplicature of the cube, &c; but the last with most success of any: he is also mentioned by Proclus, as the first who compiled elements of Mathematics.

Democritus excelled in Mathematics as well as physics; though none of his works in either kind are extant; the destruction of which is by some authors ascribed to Aristotle. The next in order is Plato, who not only improved geometry, but introduced it into physics, and so laid the foundation of a solid philosophy. From his school arose a crowd of mathematicians. Proclus mentions 13 of note; among whom was Leodamus, who improved the analysis first invented by Plato; Theætetus, who wrote Elements; and Archytas, who has the credit of being the first that applied Mathematics to use in life. These were succeeded by Neocles and Theon, the last of whom contributed to the elements. Eudoxus excelled in arithmetic and geometry, and was the first founder of a system of astronomy. Menechmus invented the conic sections, and Theudius and Hermotimus improved the elements.

For Aristotle, his works are so stored witb Mathematics, that Blancanus compiled a whole book of them: o<*>t of his school came Eudemus and Theophrastus; the first of whom wrote upon numbers, geometry, and invisible lines; and the latter composed a mathematical history. To Aristeus, Isidorus, and Hypsicles, we owe the books of Solids; which, with the other books of Elements, were improved, collected, and methodised by Euclid, who died 284 years before the birth of Christ.

A hundred years after Euclid, came Eratosthenes and Archimedes: and contemporary with the latter was Conon, a geometrician and astronomer. Soon after came Apollonius Pergæus; whose excellent conics are still extant. To him are also ascribed the 14th and 15th books of Euclid, and which, it is said, were contracted by Hypsicles. Hipparchus and Menelaus wrote on the subtenses of the arcs in a circle; and the latter also on spherical triangles. Theodosius's 3 books of Spherics are still extant. And all these, Menelaus excepted, lived before Christ.

Seventy years after Christ, was born Ptolomy of Alexandria; a good geometrician, and the prince of astronomers: to him succeeded the philosopher Plutarch, some of whose Mathematical problems are still extant. After him came Eutocius, who commented on Archimedes, and occasionally mentions the inventions of Philo, Diocles, Nicomedes, Sporus, and Heron, on the duplicature of the cube. To Ctesebes of Alexandria we are indebted for pumps; and Geminus, who lived soon after, is preferred by Proclus to Euclid himself.

Diophantus of Alexandria was a great master of numbers, and the first Greek writer on Algebra. Among others of the Ancients, Nicomachus is celebrated for his arithmetical, geometrical, and musical works: Serenus, for his books on the section of the cylinder; Proclus, for his commentaries on Euclid; and Theon has the credit among some, of being author of the books of elements ascribed to Euclid. The last to be named among the Ancients, is Pappus of Alexandria, who flourished about the year of Christ 400, and is justly celebrated for his books of Mathematical collections, still extant.

Mathematics are commonly distinguished into Speculative and Practical, Pure and Mixed.

Speculative Mathematics, is that which barely contemplates the properties of things: and

Practical Mathematics, that which applies the knowledge of those properties to some uses in life.

Pure Mathematics is that branch which considers quantity abstractedly, and without any relation to matter or bodies.

Mixed Mathematics considers quantity as subsisting in material being; for instance, length in a pole, depth in a river, height in a tower, &c.

Pure Mathematics, again, either considers quantity as discrete, and so computable, as arithmetic; or as concrete, and so measureable, as geometry.

Mixed Mathematics are very extensive, and are distinguished by various names, according to the different subjects it considers, and the different views in which it is taken; such as Astronomy, Geography, Optics, Hydrostatics, Navigation, &c, &c.

Pure Mathematics has one peculiar advantage, that it occasions no contests among wrangling disputants, as happens in other branches of knowledge: and the| reason is, because the definitions of the terms are premised, and every person that reads a proposition has the same idea of every part of it. Hence it is easy to put an end to all mathematical controversies, by shewing, either that our adversary has not stuck to his definitions, or has not laid down true premises, or else that he has drawn false conclusions from true principles; and in case we are not able to do either of these, we must acknowledge the truth of what he has proved.

It is true, that in mixed Mathematics, where we reason mathematically upon physical subjects, such just definitions cannot be given as in geometry: we must therefore be content with descriptions; which will be of the same use as definitions, provided we be consistent with ourselves, and always mean the same thing by those terms we have once explained.

Dr. Barrow gives a very elegant description of the excellence and usefulness of mathematical knowledge, in his inaugural oration, upon being appointed Professor of Mathematics at Cambridge. The Mathematics, he observes, effectually exercise, not vainly delude, nor vexatiously torment studious minds with obscure subtilties, but plainly demonstrate every thing within their reach, draw certain conclusions, instruct by profitable rules, and unfold pleasant questions. These disciplines likewise enure and corroborate the mind to a constant diligence in study; they wholly deliver us from a credulous simplicity, most strongly fortify us against the vanity of scepticism, effectually restrain us from a rash presumption, most easily incline us to a due assent, and perfectly subject us to the government of right reason. While the mind is abstracted and elevated from sensible matter, distinctly views pure forms, conceives the beauty of ideas, and investigates the harmony of proportions; the manners themselves are sensibly corrected and improved, the affections composed and rectified, the fancy calmed and settled, and the understanding raised and excited to more divine contemplations.