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

Proposition XXII.

THe Powers of Proportionals whether continuedly or discretely, such as the Squares, Cubes, &c. are also Proportional.

Demonstration.

Continual Proportionals.	Discrete Proportionals.
a ea e{powerof2}a e{powerof3}a	a ea b eb
Squares aa e{powerof2}a{powerof2} e{powerof4}a{powerof2} e{powerof6}a{powerof2}	a{powerof2} e{powerof2}a{powerof2} b{powerof2} e{powerof2}b{powerof2}
Cubes a{powerof3} e{powerof3}a{powerof3} e{powerof6}a{powerof3} e••a{powerof3}	a{powerof3} e{powerof3}a{powerof3} b{powerof3} e{powerof3}b{powerof3}

Q E. D.

SCHOLIUM.

YOu founded in this Truth, 1. the Reason of the Multipli∣cation and Division of Surd Quantities: For since from the Nature and Definition of Multiplication, it is certain, that 1 is to the Multiplier as the Multiplicand to the Product (for the multiplicand being added as many times to it self as there are Units in the Multiplier, makes the Product) if the √5 is to be multiplied by √3, then as 1 to the √3, so the √5 to the Product; and, by the present Proposition, as 1 to 3, so 5 to the □ Product, i. e. to 15. Wherefore the Product is √15; and so the Rule for Multiplying Surd Quantities is this: Multiply the Quantity under the Radical Signs, and prefix a Radical Sign to the Product.(α) 1.1 Likewise since it is certain from the Nature of Division, that the Divisor is to the Dividend as 1 to the Quotient (for the Quotient expresses by its Units how many times the Divisor is contained in the Dividend) if the √15 is to be divided by √5, you'l have √5 to the √15 as 1 to the Quo∣tient, and, by the present Scholium, 5 to 15, as 1 to the □ of the Quotient, i. e, to 3. Therefore the Quotient is the Root of 3, and so the Rule of dividing Surd Quantities this; viz.

Divide the Quantities themselves under the Radical Signs, and pre∣fix the Radical Sign to the Quotient.

II. Hence also flows the usual Reduction in the Arithme∣tick of Surds, of Surd Quantities to others partly Rational, and on the contrary, of those to the form of Surds, e. g. If you would reduce this mixt Quantity 2a√b, i. e. 2a multiplied by the √b, to the form of a Surd Quantity; which shall all be con∣tained under a Radical Sign; The Square of a Rational Quan∣tity without a Sign 4aa, if it be put under a Radical Sign, in this form √4aa, it equivalent to the Rational Quantity 2a; but the √4aa being multiplied by √b makes √4aab. by N•• 1. of this Scholium. Therefore √4aab is also equivalent to the Quantity first proposed 2a√b. Reciprocally therefore, if th•• form of a meer Surd Quantity √4aab, is to be reduced to on•• more Simple, which may contain without the Radical Sig•• whatever is therein Rational, by dividing the Quantity com∣prehended under the sign √ by some Square or Cube, &c. as here by 4aa, (i. e. √4aab by √4aa, i. e. 2a) the Quotient wil•• be √b, which multiplied by the Divisor 2a, will rightly ex∣press the proposed Quantity under this more simple Form 2a√•• Which may also serve further to illustrate the Scholia of Prop. 7. and 10.

Notes

• (α) 1.1

  Eucl. lib. 6. prop. 22.