PHILOLOGICO-MATHEMATICAL MISCELLANIE S. T HE Bulk of this Work confift- Mathema- ones, fhould make any Part thereof: But fince at this Time no Parts of Learning are found more useful, or cultivated more univerfally, or afford greater Pleasure in the Study thereof, than the Mathematical Arts and Difciplines, I thought it would be no unacceptable Service, even to the mere Philological Reader, to give fome general Account of them in a defcriptive Manner only; and that may, in fome measure, be esteem'd of a Philological Nature, and therefore comportant enough with my Defign. MATHESIS, though it originally fignifies Mathes, Learning in general, yet with us, in our own what. Tongue, it is appropriated to Mathematical Literature, and comprehends in its Signification all the Arts and Sciences which are converfant about Number, Magnitude, Meafure, and Motion, &c. They are therefore call'd Matbematical, and thofe who The Divifion who understand or profefs them are call'd Mathematicians. MATHEMATICS, or the Mathematical of Mathema- Sciences, have by many been divided into (1.) tical Arts and Pure Mathematics, containing Arithmetic and Sciences. Another Divifion thereof. Of Arithmetic. Geometry, which treat only of Number and Magnitude, and their various Habitudes and Relations abstractedly confider'd from all kind of Matter. (2.) Mixed Mathematics, which are those Branches of the Science which treat of the Properties of Quantity, either of Number or Magnitude, applied to Matter; as Aftronomy, Geography, &c. (3.) Speculative Mathematics, which contemplates the Properties, Proportions, Relations, &c. of Bodies, which make the Theory. And, (4.) Practical Mathematics, which is the Application of the Theory to the practical Ufes of Life in all the feveral Sciences. BUT this is far from being a fimple, just, and logical Divifion of the Body of Mathematical Science, I hall therefore, with regard to the particular and different Nature of the Parts, make another fourfold Divifion thereof under the general, Heads following: (1.) Arithmetic. (2.) Geometry (3) Mixed Mathematics. And, (4) Mechanics. Of all which, and their various Subdivifions, a little. ARITHMETI C. ARITHMETIC is the Doctrine of Computation in general, or the Art of eftimating Quantities of Number or Magnitude, and expreffing them in Characters of a known and determinate Value or Signification: The fundamental Rules of doing which are (after learning the Value of the Characters, which is call'd Na 6-314 Meration) PHILOLOGICO-MATHEMATICAL MISCELLANIE S. T GHE Bulk of this Work confift- Mathema- ones, should make any Part thereof; But fince at this Time no Parts of Learning are found more useful, or cultivated more univerfally, or afford greater Pleasure in the Study thereof, than the Mathematical Arts and Difciplines, I thought it would be no unacceptable Service, even to the mere Philological Reader, to give fome general Account of them in a defcriptive Manner only; and that may, in fome measure, be esteem'd of a Philological Nature, and therefore comportant enough with my Defign. MATHESIS, though it originally fignifies Mathefis, Learning in general, yet with us, in our own what. Tongue, it is appropriated to Mathematical Literature, and comprehends in its Signification all the Arts and Sciences which are converfant about Number, Magnitude, Meafure, and Motion, &c. They are therefore call'd Mathematical, and thofe who The Divifion of Mathematical Arts and Sciences. Another Divifion thereof. Of Arithmetic. who understand or profefs them are call'd Mathematicians. MATHEMATICS, or the Mathematical Sciences, have by many been divided into (1.) Pure Mathematics, containing Arithmetic and Geometry, which treat only of Number and Magnitude, and their various Habitudes and Relations abstractedly confider'd from all kind of Matter. (2.) Mixed Mathematics, which are those Branches of the Science which treat of the Properties of Quantity, either of Number or Magnitude, applied to Matter; as Aftronomy, Geography, &c. (3.) Speculative Mathematics, which contemplates the Properties, Proportions, Relations, &c. of Bodies, which make the Theory. And, (4.) Prafical Mathematics, which is the Application of the Theory to the practical Ufes of Life in all the feveral Sciences. But this is far from being a fimple, just, and logical Divifion of the Body of Mathematical Science; I fhall therefore, with regard to the particular and different Nature of the Parts, make another fourfold Divifion thereof under the general Heads following: (1.) Arithmetic. (2.) Geometry. (3.) Mixed Mathematics. And, 4) Mechanics. Of all which, and their various Subdivifions, a little. ARITHMETIC. ARITHMETIC is the Doctrine of Computation in general, or the Art of eftimating Quantities of Number or Magnitude, and expreffing them in Characters of a known and determinate Value or Signification: The fundsmental Rules of doing which are (after learning the Value of the Characters, which is call'd Ns Logarithms is this, That for every Addition and Subtraction of them, there correfponds a Multiplication and Divifion of the Number to which they belong; thus by adding 2 and 4 you have 6, which is the Logarithm of 64, the Product of 4 times 16; and the contrary for Divifion. Alfo by dividing a Logarithm you extract the Root of its Number; fo 6 divided by 2 quotes 3 the Logarithm of 8, which is the Square Root of 64; divide 6 by 3 it quotes 2 the Logarithm of 4, the Cube Root of 64; and fo of others. Having therefore a Table or Canon of Logarithms for all large Numbers, the tedious Labour of Multiplication, Divifion, and Extraction of Roots is faved by an eafy Addition, Substraction, and Divifion of their Logarithms; which is the great Excellency of this Kind of Arithmetic, and which no Artist ought, to be unacquainted with, ALGEBRA is call'd Specious Arithmetic, Of Algebra because it is a peculiar Art or Method of managing a Calculus or Computation by Species or Symbols (which are generally Letters) instead of Figures, or the Digits ufed in Common Arithmetic. The peculiar Artifice of this Invention is to affume Quantities really unknown as really known, and then to proceed with them according to the Rules till they are brought to an Equation or Equality with others given or known at first. 'Tis ufual to reprefent known Quantities by the first Letters of the Alphabet, as a, b, c, d, &c. and the unknown ones by the laft, as x, y, z, or by Vowels, as i, o, u. Moreover, to avoid all unnecessary Use or Repetition of Words, Algebraifts have a Set of Characters which fignify how the several Quantities are affected in the Operation; the chief whereof are the following. Algebraic |