So that Mathematical Science may be confidered as beginning its career with general ideas or abftracted forms; in the farther view of which we fhall find this branch of learning poffeffed of many other fingular and exclufive advantages.

One advantage is that these general or rather univerfal ideas are immediately capable of being ascertained with a logical precifion, and conveyed by clear and adequate Definitions in a language which is the most direct

ἀποδείξεις, καὶ πολλάκις ἐκ ἴσασι τὸ ὅτι· καθάπερ οἱ τὸ καθόλα θεωρονες, πολλάκις ἔνια τῶν καθ ̓ ἕκασον ἐκ ἴσασι δι' ἀνεπισκεψίαν. Εςι δὲ ταῦτα, ὅσα ἕτερόν τι ὄντα τὴν 8σίαν, κέχρηται τοῖς εἴδεσι. Τὰ γὰρ μαθήματα, περὶ εἴδη ἐσίν' ε γὰρ καθ' ὑποκειμένα τινὸς, εἰ γὰρ καὶ καθ ̓ ὑποκει μένα τινὸς, τὰ γεωμετρικά ἐσιν' ἀλλ' ἐχ ᾗ γεωμετρικά, καθ' ὑποκειμένη. Ariftot. Analyt. Poft. lib. i. cap. 13.

Linearum rectarum et circulorum defcriptiones, in quibus Geometria fundatur, ad Mechanicam pertinet. Has lineas describere Geometria non docet. Poftulat enim ut tiro eafdem accurate defcribere prius dedifcerit quam limen attingat Geometria; dein quomodo per has operationes problemata folvuntur, docet; rectas et circulos defcribere problemata funt, fed non Geometrica, ex Mechanica, poftulatur horum folutio, in Geometria docetur folutionum ufus: at gloriatur Geometria quod tam paucis principiis aliunde petitis tam multa præftat. Newtoni Præf, in Princip.

and

and obvious. Geometry defines a Point, Line, Angle, Triangle, Circle, and any other mode of continuous Quantity, the less general by the more general, in terms which are appropriate, and poffeffed of all poffible accuracy and precifion; fo that, if they be once understood, the ideas they reprefent cannot poffibly be mifconceived. And, whatever number of Units or Monades conftitute any idea of Quantity discrete (and these ideas are innumerable,) by the admirable dexterity and addrefs of the Arithmetician in the arrangement of Numbers into stated claffes and collections, general and lefs general, formed out of each other and diftinguished by appropriate names as they rife into higher and more complex orders, Tens, Hundreds, Thousands, and fo on, (an invention entitled to the gratitude of all ages and countries) its language is at once definitive, and its ideas, however complex and collective, when thus expreffed, are equally incapable of mifapprehenfion. Thus, if of Thousands we take one, of Hundreds feven, of Tens eight, and of Units nine, we have at once an adequate Defi

G 3

Definition of the idea, or collective number, of the years of the christian æra.

Another advantage fimilar to this, and by which it is heightened and completed, is, that its ideas fo adequately and easily defined, are capable of being exhibited and prefented to the eye in an obvious external shape. The diagram of a Square, Circle, or other Figure, though it cannot be a complete reprefentation of the idea, is fufficient to convey the definition through the fight directly into the understanding: And the figns of Number, which we call figures, with the order in which they are fet down 1789, form a clear and exact representation which puts the mind in immediate poffeffion of the full force of the definition- an invention which we owe to our more modern intercourfe with the east, and which the ancient mathematicians, though they had formed fome ufeful arrangements of Numbers, did not enjoy. This artifice or mechanifm of expreffion addreffed to the fight, which is the readiest and most familiar interpreter to the mind, or even to the touch, (for the great

Sanderfon

Sanderson is faid to have been born blind,) gives a superior ease and perfpicuity to Mathematics through all the ftages and progreffions of that luminous science.

So that MATHEMATICS poffefs an extraordinary clearness and precifion both in their ideas, and in their language.

Into whatever extent or variety these ideas may run, whether through all the forms and constructions of Figure, or through all the claffes and combinations of Number, and however complex and multiform they become, they are only different modifications of one and the fame kind, or, as Mr. Locke chooses to express it, of the fame idea, without the mixture or addition of any other; on which account he has diftinguished them by the name of Simple Modes, a distinction which, however expressed, is very philofophically made. They are formed by adding unit to unit and line to line through all the modifications of Number and Figure, without the mixture of any thing from which circumstance the fcience

else;

in queftion derives this great and exclusive privilege, that its ideas are totally feparate and diftinct from thofe of every other kind.

And, however numerous they may be, another advantage to the precifion of the fcience to which they belong, is, that every one is abfolute and unchangeable in itself, that is, it cannot be either greater or lefs," or any way different from what it exactly is, by partaking or communicating with any other even of the fame kind; for two numbers differing only by one unit, or two angles by one degree, are as abfolutely different from each other as those that are the most diftant." So that Mathematical ideas are individually diftinct from one another, as well as totally from those of other kinds.

They have, therefore, only to do with themselves, at the fame time that they stand

Quantitas non recipit majus aut minus. See Ariftot. Categ. cap. vi.

Two is as different and diftinct from one as from a thousand; but ideas of good and evil, hot and cold, hard and foft, and of the different colours, participate with each other, and are more or lefs akin, varying into fhades compounded of their neighbours, and having their difference according to their distance.

perfectly