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

CAAP. II. Of the Powers of QUANTITIES.

Containing (after a compendious Way) most part of the 2d Book of Euclid; and the Appendix of Clavius to Lib. 9. Prop. 14.

Proposition VI.

IF any whole Quantity be divided into two parts(α) 1.1 the Rectangle contained under the whole, and one of its parts, is equal to the Square of the same part, and the Rectangle contain'd under both the parts.

Demonstration.

Let	a+b	represent the whole	a+b
	b	one part of it, or	a	the other.
	ab+bb	the Rectangle,	aa+ab	the Rectangle.

(See Fig. 50.) Q. E. D.

Proposition VII.

IF a whole Quantity be divided into two parts(β) 1.2 the Square of the whole is equal to the Squares of both those parts and 2 Rectangles contained under them.

Demonstration.

This is evident from the preceding, and may moreover thus appear further.

Let the Parts be a and b, then will the whole be 〈 math 〉〈 math 〉

Which if you multiply by it self 〈 math 〉〈 math 〉

You have the Square 〈 math 〉〈 math 〉

(See Fig. 51. N. 1.) Q.E.D.

CONSECTARYS.

I. HEnce you have the Original Rule for Extracting of Square Roots, as we have shewn after Definition 30. and here have further Illustrated in Scheme N^o 2.

II. Hence it naturally follows, that the Square of double any Side is Quadruple of the Square of that Side taken singly.

III. Hence also you have the addition of surd Numbers, or in general of surd Quantities, by help of the following Rule (supposing in the mean while their Multiplication:) Suppose these 2 Surds √8 and √18, or more generally √75aa and √27aa, are to be added together; first add their Squares 8 and 18, &c. then double their Rectangle (√144) that is, multiply it by the √4, and then the double of this √576, i. e. having ex∣tracted the Square Root, (24) and added it to the Sum of the first Squares (26) the Root of the whole Summ (50) viz. √50, is the Sum of the two surd Quantities first proposed.

SCHOLIUM.

BUT if it happens that the Root of the double Product can∣not be expressed by a Rational Number (as, when the proposed Quantities are Surds, as √3 and √7, to whose Squares 3+7, i. e. 10, you must add the double Product of √7 by √3, i. e. √84, which cannot be expressed by a Rational Num∣ber) then that double Product must be joined under a Surd Form, or Radical Sign, to the Sum of the Squares (thus, viz. 10+√84) and to this whole Aggregate prefix another Radi∣cal Sign, thus, 〈 math 〉〈 math 〉; or also you may only simply join the Surd Quantities proposed by the Sign + thus, √3+√7. Here also you may note, that the two Surd Quantities proposed

in the first case of Consectary 2. are called Communicants; in the other case of this Scholium, Non-Communicants: For in this case each quantity under the Radical Sign may be divided by some Square, and have the same Quotient (e. g. 8 and 18, may be divided the first by 4, the other by 9, and the Quotient of both will be 2; likewise 75aa and 27aa may be divided, the one by 25aa, the other by 9aa, the Quotient of both being 3; and then if the Quotient on both sides be left under the Radical Sign, and the Root of the dividing Square set before it, the same quantities will be rightly expressed under this form: 2√2 and 3√2, also 5a√3 and 3a√3; and then the addition is easie, viz. only collecting or adding together the Quantities prefixt to the Radical Sign; so that the Sums will be of the one 5√2 and of the other 8a√3, which are indeed the same we have shewn in Consect. 2. For if contrarywise we square the Quantities that stand without, or are prefixt to the Radical Sign, and then set those Squares (25 and 64aa) under the Ra∣dical Sign, multiplying by the Number prefixt to it, you'l have for the one √50, for the other √192aa (Consect. after Schol-Prop. 22.)

Proposition VIII.

IF any whole Quantity (viz. Line or Number) be divided(α) 1.3 into two equal parts, and two unequal ones, the Rectangle of the unequal ones, toge∣ther with the Square of (the intermediate part or) the difference of the equal part from the unequal one, is equal the Square of the half.

An Universal Demonstration.

Suppose the parts to be a and a, and the whole 2a; let one of the unequal Parts be b, the other will be 2a−b, and the dif∣ference between the equal and unequal part a−b.

The equal ones 〈 math 〉〈 math 〉

Rectangle 〈 math 〉〈 math 〉

Difference 〈 math 〉〈 math 〉

The Sum will be aa (the other parts destroying one another) Q.E.D. (Vid. Fig. 52.)

Proposition IX.

IF to any whole Quantity divided into two equal parts(α) 1.4 you add another Quantity of the same kind, the Rect∣angle or Product made of the whole and the part added, mul∣tiplied by that part added, together with th•• square of the half, will be equal to the Square o•• the Quantity compounded of that half, and th•• part added.

Demonstration.

Let the whole be called 2 a, the part added b, then th•• quantity compounded of the whole and the part added will b•• 2a+b; and that compounded of the half and the part added a+b.

The Quantity compounded of the whole, and the part added is, 〈 math 〉〈 math 〉 the half a Comp. 〈 math 〉〈 math 〉

Multip. by the part added 〈 math 〉〈 math 〉

2ab+bb□ of the half aa=□aa+2ab+bb

(Vid. Fig. 53.) Q. E. D.

Proposition X.

IF a Quantity be divided any how into(b) 1.5 two parts, the Square of the whole, together with that of one of its parts, is equal to two Rectangles contained under the whole and the first part, together with the Square of the other part.

[illustration]

Pag. 63.

54

55

56

57

58

59

60

The Universal Demonstration.

Let a be one part and b the other, the whole a+b

a+b	the whole.	The whole	a+b
a	the first part *		a+b
aa+ab			aa+ab
2			ab+bb
2aa+2ab	the double rectangle	□ of the whole	aa+2ab+bb
add bb	the □ of the other part	* add	aa
Sum 2aa+2ab+bb	= to the Sum	....	2aa+2ab+bb

(Vid. Fig. 54. N^o 1.) Q.E.D.

CONSECTARY.

HEnce you have the Subtraction of Surd Numbers, or more generally of Surd Quantities, by help of the following Rule.

Add the Squares of the given Roots according to Consect. 3. Prop. 7. and from their Sum subtract the double Rectangle of their Roots; the Root of the Remainder will be the difference sought of the given Quantities.

As, if the √8(BC) is to be subtracted from √50AC (Fig. 54, N^o 1.) you must add 50, i. e. the whole Square AD) and 8, (i. e. the other Square superadded DE,) and the Sum will be 58, equal to the two Rectangles AF and FE+□GH, by ••his Prop. I find therefore those two Rectangles by multiplying √50 by √8, and then the Product √400 by 2 or √4, there∣by to obtain the double Rectangle √1600, i. e. (having actu∣ally Extracted the Root) 40. This double Rectangle therefore ••or 40, being subtracted out of the superiour Sum, the remain∣der 18 will be the □ GH, and so its Root (viz. √18) gives ••he required Difference between the given Surd Quantities.

SCHOLION.

BUT this Subtraction may be performed yet a shorter way, if each quantity under its Radical can be divided by some

square, so that the same Quotient may come out on both sides that is, if the Surd Quantities are Communicants, as e. g. √5•• (the number 50 being divided by 25) is equal to 5√2 a•• √8 to 2√2; for then the numbers prefixt to the Radical Sig•• being subtracted from one another (viz. 2√2 from 5√2) yo•• have immediately the remainder or difference 3√2, i. e. √1•• But if the proposed Quantities are not Communicants (as if th•• √3 is to be subtracted from √7) the remainder may be brief•• expressed by means of the Sign − thus, 〈 math 〉〈 math 〉, or accordin•• to the foregoing Consectary, thus, 〈 math 〉〈 math 〉.

Proposition XI.

IF any Quantity be divided into two parts,(α) 1.6 the Quadru•• Rectangle contained under the whole and one of its parts, toget•••••• with the Square of its other parts, will be equal to the Square of 〈◊〉〈◊〉 Quantity compounded of the whole and the other part.

Demonstration.

Suppose	a+b	the whole.
	b	one part.
	ab+bb	the Rectangle of these two.
mult. by	4
	4ab+4bb	the Quadruple Rectangle.
Add	aa	the Square of the other part.
Sum	aa+4ab+bb

The Quantity compounded of the whole and the first part 〈 math 〉〈 math 〉

Square of the Compound Quantity Q. E ••

(Vid. Fig. 55.)

Proposition XII.

IF any Quantity be divided into two equal parts(β) 1.7 and into two other unequal ones, the Squares of the unequal parts taken together will be double the Square of half the quantity, and the Square of the difference, viz. of the equal and unequal part, taken to∣gether.

Demonstration.

Suppose the equal parts to be a and a, the difference (b) the greater of the unequal Parts to be a+b, the less a−b.

The greater part 〈 math 〉〈 math 〉	The less 〈 math 〉〈 math 〉	Half 〈 math 〉〈 math 〉	Difference 〈 math 〉〈 math 〉
Sum of these 2aa+2abb	Sum aa+bb

Q. E. D. (Vid. Fig. 56.)

Proposition XIII.

IF to any whole Quantity(α) 1.8 divided into two equal parts there be added another Quantity of the same kind, the Square of the Quantity compounded of the whole, and the quantity added, together with the square of the quantity added, will be double the square of the half the quantity, and the square of the Sum of the half and the part added taken together.

Demonstration.

Suppose the whole to be 2 a, the half parts a and a, the quantity added b; then the quantity compounded of the whole and the quantity added, will be 2a+b, and that of the half and the quantity added a+b.

Comp. of the whole 〈 math 〉〈 math 〉 and quantity added

Sum

Half 〈 math 〉〈 math 〉

Qu. compouded of hal•• 〈 math 〉〈 math 〉 and qu. added,

Sum 2aa+2ab+•••• Manifestly the half of the for¦mer Sum. Q.E.D.

Notes

• (α) 1.1

  Eucl. lib. 2. prop. 3. A. C. also the third.
• (β) 1.2

  Eucl. Prop. 4. lib. 2.
• (α) 1.3

  Eucl. & Clav. 5.
• (α) 1.4

  Eucl. & Clav. 6.
• (b) 1.5

  Eucl. & Clav. prop. 7. l. 2
• (α) 1.6

  Eucl. & Clav. prop. 8.
• (β) 1.7

  Eucl & Clav. prop. 9.
• (α) 1.8

  Eucl. & Clav. x.