Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

DEFINITION XXVIII.

MƲltiplication, generally Speaking, is nothing else but a Complex or manifold Addition of the same quantity, wherein that which is produced is peculiarly call'd the Product, and those quantities by which it is produced, are called the Multiplicand and the Multiplier: The first denotes the Quantity which is to be multiplied, or added so many times to its self; and the other the Number by which it is to be multiplied, or determins how many times it is to be added to it self. The same terms are applyed moreover to Lines and other Quan∣tities. But here are two things to be chiefly noted; 1. That the Multiplication of one number by another, or of a Line by a Line, may be considered as having a double Event; for the Product may be either of the same or a different kind, as, e. g. when . . . . 4 is multiplied by 3 . . . the product may be considered either as a Line, thus, . . . . . . . . . . . . or as

a Plane Surface in this Form,

[illustration] 12 dots arranged in three rows and four columns

Whence it is also named a Plane Number, and the product is conceived to be formed by the motion of an erect Line AB, consisting of 3 equal parts, along another BC, consisting of 4 equal parts, and conceived as lying along. So also the Multiplication of Lines (e. g. of the Line A — B by the Line B—C) may be conceived to be so performed, that the Product also shall be a Line, e. g. C—D (concerning the usefulness of which Multiplication in Geometry, we shall have occasion to speak more hereafter;) or so, that the Pro∣duct shall be a Plane or Surface, arising from the motion of the erect Line AB, along AC, conceiv'd as lying along; as we have already shewn. But as for the most part these Planes so produced are called Rectangles, if the Lines that form them are unequal; but if they are equal they are call'd Squares, (otherwise the Powers of the given Quantities;) and in this case the Lines that form them are called Square Roots; so also if those Planes are multiplied again into a third Quantity (as either a Line or a Number) there will arise Solids, and parti∣cularly if that third Quantity be the Root of the Square, the Product is called a Cube, &c. The other thing to be noted is, That both these ways of Multiplying either Numbers or Lines, are expressed by a very compendious, tho arbitrary way, of Notation, viz. by a bare Juxtaposition of the Letters which denote such and such Species of Quantities, as, e. g. if for the forementioned Number or Line AB we put a, and for BC b, the Product will be ab; or if the Efficients are equal, as a and a the Square thence produced, will be aa or a{powerof2}; and if this Square be further multiplied by its Root a, then the Cube thence produced will be aaa or a{powerof3}, &c. Which being premised, you have these following

CONSECTARIES.

I. IF a Positive Quantity be multiplied by a Positive one, the Product will be also Positive; since to multiply is to repeat the Quantity according as the Multiplier directs: Where∣fore to multiply by a Positive Quantity, is to repeat the Quan∣tities positively; as on the other side, to multiply by

a Privative, is so many times to repeat the Privation of that Thing: Which we shall shew further hereafter.

II. Equal Quantities (a and a) multiplied by the same (b), or contrariwise, will give equal Products (ab and ab or ba).

III. The same Quantity (z) multiplied by the whole Quan∣tity (a+b+c) or by(a) 1.1 all its parts separately, will give equal Products. Also

IV. The whole (a+b) whether it be mul∣tiplied by(b) 1.2 it self, or by its parts separately, will give equal Products.

SCHOLIUM I.

THe Vulgar Praxis of Numeral Multiplication, is founded on these two last Consectarys, as e. g. to multiply 126 by 3; you first multiply 6 by 3, then 2, i. e. 20 by 3, then 1, i. e. 100 by the same, and then add each of those partial Products into one Sum: In like manner being to multiply 348 by 23, you first multiply each Note of the Multiplicand by the first of the Multiplier (3) and then by the second (2) (i. e. 20) &c. which is to be done likewise after the same man∣ner in Tetractical Multiplication; only in this latter, which is more easie, you have nothing to reserve in your mind, but all is immediately writ down, (which might also be done in Vulgar Multiplication) as may be seen by this Example un∣derneath, as also the great easiness of this sort of Multiplica∣tion, beyond the common way, because there is no need of any longer Table than that we have shewn page 7.

〈 math 〉〈 math 〉

SCHOLIUM II.

It is manifest from what we have said,

I. IF the Base of a Parallelogram be called (b) and its Alti∣tude a, its Area may be expressed by the Product ab, by Cons. 7. Definit. 12.

II. If the Base of a ▵ be b or eb, and its Altitude a its Area will be half ab or half eab, by Consectary 8. of the same De∣finition.

III. If the Base of a Prism or Parallelepiped or Pyramid be half ab or ab, and its Altitude c, the solid Contents of that Prism will be half abc, and of the Parallelepiped abc, by Con∣sect. 3 & 4. Def. 16. and of the Pyramid ⅙ abc, by Cons. 3. Def. 17.

Notes

• (a) 1.1

  Eucl. lib. 2. prop. 1.
• (b) 1.2

  lib. 2. prop. 2.