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

DEFINITION XXVII.

SUbtraction is the taking one Quantity from another (of th•• same kind;) which is so performed that either the remainde obtains a new Name, or by a bare separation of the Subtrahen•• by the privative Particle less, or the usual Sign − which stand for it, as e. g. . . . or three being subtracted from . . . . . . ▪ or 7, the remainder or difference is . . . . or 4 and this Lin•• — Subtracted from that — leaves — Now if we would signify this more generally either of the•• Lines, or the Number above, or any 2 Quantities whatsoeve•• that are to be Subtracted one from the other, by naming th•• first (a) and the latter (b) we shall have the remainder a — •• Herein are evident these and the like Axioms: If from equ•••• Quantities you Subtract Equal ones, the Remainders or Differences 〈◊〉〈◊〉 be equal. Here it will be worth while to take notice of, from this and the preced. Definit. the following

CONSECTARYS.

I. IF a negative Quantity be added to it self considered a positive (as − 3 to + 3 or − a to + a) the Sum wi•••• be 〈◊〉〈◊〉 for to add a Privation or Negative is the same thing a•• to Subtract a Positive, wherefore to join a Negative and Pos••∣••••ve together, is to make the one to destroy the other.

II. If a negative be subtracted from its positive (−a from +a) the remainder will be double of that positive (+2a) for to subtract or take away a privation or negative, is to add that very thing, the privation of which you take away; for really that which in words is called the addition of a Privation, is in reality a Subtraction, and a subtraction of it, is really an ad∣dition; and what is here call'd a Remainder, is indeed a Sum or Aggregate; and what is there call'd a Sum, is truly a Remain∣der. Thus,

III. If the positive Quantity (+a) be taken from the pri∣vative one (−a) the remainder is double the privative one (−2a) since, taking away a positive one, there necessarily arises a new Privation which will double that you had before. Hence,

IV. You have the Original of the Vulgar Rules in Literal Addition and Subtraction: If the Signs of the unequal Quantities are different, in the room of Addition you must subtract, and in room of Subtraction add, and to the sum or remainder, prefix the Sign in the first place of the greatest, in the next of that from which you Subtract: but if the Signs are both the same, and the greatest quan∣tity to be subtracted from the less, you must, on the contrary, subtract according to the natural Way, the least from the greater, and prefix the contrary Sign to the remainder: Which Rules you may see Illustrated in the following Examples:

Addition	Subtraction.
4b−2a	from 2a+b	from 3a+2b
3b+5a	Subst. a−b	Subst. 2a+3b
7b+3a	R. a+2b	R. a−b

NOTE.

☞ Instead of the Authors 4th Consect. as far as it relates to Subtraction, which may seem a little perplext, take this ge∣neral Rule for Subtraction in Species, viz. Change all the Signs of the lower Line, or Subtrahend, and then add the Quantities, and you have the true Remainder.

SCHOLIUM.

IN this Literal Subtraction, we have not that conveniency which the invention of Vulgar Notes supplies us with, that from the next foregoing Note we may borrow Ʋnity, which in the following Series goes for 10, &c. This is done in Te∣tractycal Subtraction only with this difference, that an Unite here borrowed goes only for 4. That the easiness of this O∣peration may appear, we will add one Example, wherein from this number, — you are to subtract this,

1232002310232

321012321223

310323323003

Whereever therefore the inferiour Note is greater than the superiour one, the facility is much greater here than in com∣mon Subtraction, because never a greater number than 3 is to be subtracted out of a greater, than 4 and 2: but if the in∣feriour number be greater than the superiour, you borrow unity from the left hand, which is equivalent to 4; the rest is perform'd as in common Subtraction.