CHAP. XXXI.

MATHEMATICS,
Continued.

Advantages, History and Province of Geometry-Principles of Geometry. Elementary Treatises, Simson's "Elements”—CunnTacquet-De Chales-Whiston-Barrow-Simpson's Bonnycastle's-Payne's and Cowley's Geometry. Matton's-Playfair'sLeslie's-Reynard's; and Keith's. Application of Algebra to Geometry. Simpson-Frend-Bonnycastle-Lady's Diary, and Leybourn's Mathematical Repository.

NEXT to Arithmetic should follow Geometry, in a course of liberal and scientific education. Geometry literally signifies measuring the earth, or parts thereof; and it was probably first invented to enable people to ascertain their own property in land, since which it has been extended and applied to other things and for other purposes, insomuch that geometry, with arithmetic, is now regarded as the foundation of all mathematics.

"Geometry," says an excellent writer, "will enable a person to think justly. Without it there is a certain method wanting which is necessary to rectify our thoughts, to arrange our ideas, and to determine our judgments aright. It is easy to perceive in reading a book, even a moral one, whether the author be a mathematician or not. I am seldom deceived in this observation. The famous French metaphysician would not

have composed the Inquiry after Truth,* nor the famous Leibnitz his Theodicé, if they had not been mathematicians. We perceive in their productions that geometrical order which brings their reasonings into small compass, while it gives them energy and method. Order is delightful; there is nothing in nature but what is stamped with it, and without it there could be no harmony. We may likewise say that the mathematics are an universal science, which connects all the rest, and displays them in their happiest relations. The mathematician, at the first look, is sure to analyse and unravel a subject or proposition with justness; but a man who does not understand this science, sees only in a vague, and almost always in an imperfect manner. Apply yourself then to this great branch of knowledge, so worthy of your curiosity, and so necessary to the uses of life; but not in such a degree as to throw you into absence;-endeavour to be always recollected, whatever are your studies. If I were young, and had leisure, I would acquire a more extensive knowledge of geometry. I have always cherished that science with a particular predilection. My turn of mind made me seek with avidity every thing that was methodical; and I pay but little respect to those works which are only the exercises of imagination. We have three principal sciences, which I compare to the three essential parts of the human composition :-Theology, which, by its spirituality, resembles our soul; the mathematics, which, by their combination and justness, express our reason; and natural philosophy, which, by its mechanical operations, denotes our bodies: and these three sciences (which ought to maintain a perfect harmony) while they keep within their proper sphere, necessarily elevate us towards their Author, the source and fulness of all light. Philosophy without geometry, is like medicine without chemistry. The greater number of modern philosophers reason inconclusively, only be cause they are unacquainted with geometry. They mistake

* Malebranche.

sophisms for truths; and if they lay down just principles, they deduce false conclusions from them."

Herodotus, Diodorus, and Strabo, maintain that the Egyp tians were the first inventors of geometry; and that the annual inundations of the Nile were the occasions of it; for that river, bearing away all the bounds and landmarks of men's estates, the people were obliged to distinguish their lands by the consideration of their figure and quantity, and thus formed for themselves a method or art which was the origin of geometry. A farther contemplation of the figures of lands laid down for this purpose, might probably lead them to the discovery of some of the properties of those figures; which speculation continually improving, the art also improved till it laid claim to the rank of a science.

Geometry then, may be considered as the science of extension, or extended things, that is, of lines, superficies, and solids. Notwithstanding what has been said above of the Egyptians being the inventors of Geometry, the fact has been disputed, and the honour given, by very respectable authors, to the Hebrews. There is, however, no doubt that the inhabitants of Egypt, in their ancient monarchical state, were acquainted with the elements of geometry, though it does not appear that they had gone deeply into it, since to Pythagoras, who flourished about five hundred and twenty years before the birth of Christ, and who had spent a considerable part of his life in Egypt, was attributed the invention of certain propositions in Euclid, particularly the 47th of the first book, which is called after his name, the " Pythagorean theorem," and for the discovery of which he offered a hecatomb to the gods. Hence it has been inferred, that the great learning of the Egyptians was not geometrical.

From Egypt, geometry, probably in its infant state, passed over into Greece; for Thales, the Milesian, who flourished. five hundred and eighty-four years before Christ, was reported to be the first of the Greeks, who, coming into Egypt, transferred Geometry from thence into Greece. He is said to

have discovered several of the propositions of the first five books of the Elements which go under the name of Euclid. After Thales, came Pythagoras, already cited, who first of all abstracted geometry from matter, and made many discoveries. Next flourished Anaxagoras, Hippocrates, and many others, till we come to Plato, than whom no one shed a greater lustre on the mathematical sciences; he made many considerable additions to geometry, and upon the entrance to his academy, the inscription "Let no one unacquainted with geometry enter," was written. The fifth book of the Elements is said to have been the production of Eudoxus; and to Aristeus, Isidore, and Hypsicles, we are indebted for the books of the solid geometry. After these Euclid came, who collected the inventions and discoveries of others, disposed them into order, in many respects improved them, and left those Elements, by which, in some shape or other, the youth of every succeeding generation, from that time to this, have been instructed in mathematics. Euclid died about two hundred and eighty-four years before the birth of Christ.

The next to Euclid of the ancient writers, whose works are extant, is Apollonius Pergeus, who flourished in the time of Ptolemy Euergetes, about a century later than Euclid. The third ancient Geometer, whose writings remain, is Archimedes of Syracuse, who was celebrated at the same time with Apollonius; to the works of this great man we shall have occasion again to refer. We might mention many other names of great celebrity among the Greeks, which have been immortalized by their skill in ancient Geometry. This people continued their attention to the sciences, properly so called, even Whereas after they had been subdued by the Romans. their conquerors were so little acquainted with this science, even in the most flourishing time of their republic, that they commonly gave the name of mathematicians to those who pursued the chimeras of judicial astrology. Nor were they more disposed to cultivate geometry, it will be readily imagined, during the decline, and after the fall of the Roman empire. The case was different with the Greeks, among

whom we find many excellent geometers since the commencement of the Christian era, and even after the translation of the Roman empire, Ptolemy lived under Marcus Aurelius; and we are in possession of the works of Pappus of Alexandria, who flourished in the time of Theodosius; of the Commentary of Eutocius the Ascalonite, who lived in the middle of the sixth century, on Archimedes' mensuration of a circle; and of a Commentary on Euclid by Proclus, who flourished still later.

The inundation of ignorance and barbarism, to which we have referred in a preceding chapter, was as unfavourable to geometry as to the other sciences; and the few, who even dared to apply themselves to it, were calumniated as magicians. A gleam of light, however, soon appeared, and in those times of European darkness, the Arabians themselves, as we have seen, became distinguished as the guardians and promoters of science; and from the ninth to the fourteenth century, they produced many astronomers, geometers, geographers, &c.; from whom the mathematical sciences were again received into Spain, Italy, and the other parts of Europe, at the close of the fourteenth century. After this period, many editions of Euclid, and many commentaries on his Elements were published.

At the revival of letters, there were few Europeans capable of translating and commenting on the works of the ancient Geometers, and the science of Geometry made little progress till the time of Des Cartes, who published his Geometry in 1637; but from that period to the present, it has abounded with votaries in almost all civilized nations, but in none more than in Great Britain.

The province of Geometry is almost infinite: few of our ideas but may be represented to the imagination by lines, by means of which they become of geometrical consideration: it being geometry alone that makes comparisons, and finds the relations, of lines. Astronomy, music, mechanics, optics, and in short all the sciences which consider things susceptible of more or less, may be referred to geometry; for all speculative