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PHILOLOGICO-MATHEMATICAL

M I S CE L L A N I E S.

HE Bulk of this work consist- Mathema

ing purely of Philological Litera- tical LiteraT ture, it could not well be ex-fure, wby,

bere described. pected that Subjects of any other Sort, especially Mathematical

ones, should make any Part thereof: But since at this Time no Parts of Learning are found more useful, or cultivated more universally, or afford greater Pleasure in the Study thereof, than the Mathematical Arts and Disciplines, I thought it would be no unacceptable Service, even to the mere Philological Reader, to give some general Account of them in a descriptive Manner only; and that may, in some meafure, be esteem'd of a Philological Nature, and therefore comportant enough with my Design.

MATHESIS, though it originally signifies Mathefi, Learning in general, yet with us, in our own what. Tongue, it is appropriated to Matbematical Literature, and comprehends in its Signification all the Arts and Sciences which are conversant about Number, Magnitude, Measure, and Motion, &c. They are therefore called Matbematical, and those

who

who understand or profess them are callid Ma

thematicians. The Division

MATHEMATICS, or the Mathematical of Mathema- Sciences, have by many been divided into (1.) tical Arts and Pure Mathematics, containing Arithmetic and Sciences.

Geometry, which treat only of Number and Mag-
nitude, and their various Habitudes and Relations
abstractedly consider'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, ap-
plied to Matter ; as Astronomy, Geograpby, &c.
(3.) Speculative Mathematics, which contemplates
the Properties, Proportions, Relations, &c. of Bo-
dies, which make the Theory. And, (4.) Prakti-
cal Mathematics, which is the Application of the
Theory to the praĉtical Uses of Life in all the le-

veral Sciences.
Another . But this is far from being a simple, just, and
Division

logical Division of the Body of Mathematical Scithereof.

ence ; 1 Thall therefore, with regard to the par-
ticular and different Nature of the Parts, make
another fourfold Division thereof under the ge-
neral Heads following: (1.) Arithmetic. (2.)
Geometry. (3.) Mixed Mathematics. And, (4.)
Mechanics. Of all which, and their various Sub-
divisions, a little.

Of Arithmetic.

A'R IT HME T I C. ARITHMETIC is the Doctrine of Computation in general, or the Art of estimating Quantities of Number or Magnitude, and ex. pressing 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 callid Na

meration)

1

meration) five, viz. (1:) Addition, by which van Its Rules. rious and different Numbers of Things are col !! lected into one Sum, which is calld the Total or Amount of all. (2.) Substraction, by which one Number or Quantity is taken from another in order to know the Remainder, Difference, or Excess of the Greater above the Leffer. (3-) Multiplication, by which one Number, calid the Multiplicand, is increased or multiplied by another, callid che Multiplier, so many times as is express'd thereby; the Result of which is called the Produt. (4.) Division, by which one Number, callid the Divisór, may be substrated from anocher, call'd the Dividend, so many times as it is contain!d therein, which is express'd by a third Number call'd the Quotient. (5.) Evolution, or the Extraction of Roots out of any Power, as the Square, the Cube, the Biquadrate, the Surfolid, &c. which are produced by multiplying any Number, call'd the Root, into itself 1, 2, 3, 4, 170 5,&c. times respectively.

The Art of Computation consists of the follow. The fevral ing Branches, viz. (1.) Numerical Arithmetic, or Kinds thereof. that which performs by Numbers. (2.) Logarithmetical Arithmetic, or that which computes by Logarithms or the Ratio's of Numbers. (3.) Specious Arithmetic, or Algebra, which useth Symbols or Chara&ters instead of Numbers. And" (4:) Fluxionary, which proceeds with the momentary Increments and Decrements of Quantity conlidera in a flowing State.

Numerical Computation makes use of nine Cha- Numerical racters, calld. Figures or Digits, to express Nam- or Vulgar, bers by, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, and the Cyphero; and it is of two Kinds; viz. (1.) Vulgar or Common Arithmetic, which exprefseth the Value of Money, Weights, Measures, and Fractional

Parts

Parts in divers Denominations, according to the and Decimal Usage of the Country. (2.) Decimal Arithmetic; Arithmetic.

this exprefseth the Value of divers inferior Divifions or Parts of Money, Weight, Measure, Time, in tentb, bundredth, thousandth, &c. Parts of the whole Number or Integer ; that is, the Integer is supposed to be divided into 10, 100, 1000, 10000,&c. equal Parts; then the inferior Denominations or Parts of this Integer are express'd in those equal Parts, which, because their Value decreaseth in a tenfold Proportion in each Place to the Right-hand of the Integer, are call'd Decimal Parts or Numbers : For Instance, in Vulgar Arithmetic 121. 155. 8 d. will be thus expressed in Decimals, 12,7854, which are to be work'd in all Respects like Whole Numbers ; which therefore renders Decimal Arithmetic compendious, easy, and every way preferable to che

Vulgar. of Logarithms LOGARITHMS are Numbers in Aritb

metical Progreshon, which, fet with others in a
Geometrical Progrefron, do express their Ratio's
or Proportions to one another, as in the two fol.
lowing Series, viz.
Thus
{

Logarithms, o. 1. 2. 3. 4. 5. 6. Aritb. Pres.

Numbers, 1. 2. 4. 8. 16. 32. 64. Geom. Prog. Now the Ratio or Number by which the Members of the Geometrical Progefion are produced by a constant Multiplication therewith, is 2 ; thus 8' is produced by 3 Multiplications, 16 by 4, 32 by 5, &c. And therefore the Ratio of 8 to I (the first Term) is 3, of 16 to 1 is 4, of 32 to ì is 5,&c. All which Ratio's are expressed by the Numbers 3, 4, 5 in the Series above; for which Reason they are call'd their Logaritbms. Now the peculiar and most useful Property of

Logarithms

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