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· So that Mathematical Science may be confidered as beginning its career with general ideas or abstracted forms ;' in the farther view of which we shall find this branch of learning poffefsed of many other fingular and exclufive advantages.
One advantage is that these general or rather universal 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
αποδείξεις, και πολλάκις εκ ίσασι το ότι καθάπερ οι το καθόλα θεωρενες, πολλάκις ένια των καθ' έκασον εκ ίσασί δι' ανεπισκεψίαν. Εςι δε ταύτα, όσα έτερόν τι όντα την 8olov, xéxentas toho sillesi. Tà gão na SKWATA, wipi eidai isíve å yao xanla Únoxeluéva Tivbs, si gae xai xaf' vmOXE: jéva tuvos, så pewm.etpincé ésivo dnx'x ñ yowp.erpixã, x& J* ÚTI CHEI Jéve. Aristot. Analyt.Poft. lib. i. cap. 13.
Linearum rectarum et circulorum descriptiones, in quibus Geometria fundatur, ad Mechanicam pertinet. Has lineas describere Geometria non docet. Postulat enim úc tiro eafdem accurate describere prius dediscerit quam limen attingat Geometriæ ; dein quomodo per has operationes problemata solvuntur, docet; rectas et circulos defcribere problemata funt, fed non Geometrica, ex Mechanica, poftulatur horum solutio, in Geometria docetur folutionum usus : at gloriatur Geometria quod tam paucis principiis aliunde petitis tam multa præstat. Newtoni Præf. in Princip.
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 pofseffed of all possible accuracy and precision; so that, if they be once understood, the ideas they represent cannot possibly be misconceived. And, whatever number of Units or Monades constitute any idea of Quantity discrete (and these ideas are innumerable,) by the admirable dexterity and address of the Arithmetician in the arrangement of Numbers into stated classes and collections, general and less general, formed out of each other and distinguished by appropriate names as they rise into higher and more complex orders, Tens, Hundreds, Thousands, and so 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 expressed, are equally incapable of mifapprehension. Thus, if of Thousands we take one, of Hundreds feven, of Tens eight, and of Units nine, we have at once an adequate
Definition of the idea, or collective number, of the years of the christian æra.
Another advantage similar 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 presented to the eye in an obvious external shape. The diagram of a Square, Circle, or other Figure, though it cannot be a complete representation of the idea, is sufficient to convey the definition through the fight directly into the understanding: And the signs of Number, which we call figures, with the order in which they are set down 1789, form a clear and exact representation which puts the mind in immediate possession of the full force of the definition — an invention which we owe to our more modern intercourse with the east, and which the ancient mathematicians, though they had formed fome useful-arrangements of Numbers, did not enjoy. This artifice or mechanism of expreffion addressed to the fight, which is the readiest and most familiar interpreter to the mind, or even to the touch, (for the great
Sanderson is said to have been born blind,) gives a superior ease and perspicuity to Mathematics through all the stages and progresfions of that luminous science.
So that MATHEMATICS possess an extraordinary clearness and precision 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 classes and combinations of Number, and however complex and multiformn they become, they are only different modifications of one and the same kind, or, as Mr. Locke chooses to express it, of the same idea, without the mixture or addition of any other ; on which account he has distinguished them by the name of Simple Modes, a distinction which, however expressed, is very philosophically 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 elle; from whịch circumstance the science
in question derives this great and exclusive privilege, that its ideas are totally separate and diftin&t from those of every other kind.
And, however numerous they may be, another advantage to the precision of the fcience to which they belong, is, that every one is absolute and unchangeable in itself, that is, it cannot be either greater or less, or any way different from what it exactly is, by partaking or communicating with any other even of the same kind; for two numbers differing only by one unit, or two angles by one degree, are as absolutely different from each other as those that are the most distant." 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 same time that they stand
* Quantitas non recipit majus aut minus. See Aristot, Categ. cap. vi.
Two is as different and distinct from one as from a thousand; but ideas of good and evil, hot and cold, hard and soft, and of the different colours, participate with each other, and are more or less akin, varying into shades compounded of their neighbours, and having their difference according to their distance,