The description and use of two arithmetick instruments together with a short treatise, explaining and demonstrating the ordinary operations of arithmetick, as likewise a perpetual almanack and several useful tables : presented to His most excellent Majesty Charles II ... / by S. Morland.

CHAP. XVI. The Reason and Demonstration of the Vulgar Operation of Extracting the Square-Root.

ANd after this very manner and method may the Square-Root of any Plain Number in Integers be

extracted, though never so great; but that this and all other Operations of the same nature may be also performed with understanding, and satisfaction, it will be necessary to make some reflections upon the nature and genesis of a Square-Number, and in order there∣unto, the Practitioner is desired to consider the fol∣lowing Definition of a Square-Number.

Eucl. l. 7. Defin. XVIII. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, &c.

A Square-Number is that which is equally equal, or, which is contained under two equal Numbers.

THus the Square-Number (4) is contained under two equal Numbers, viz. (2) and (2) and the Square-Num∣ber (9) is contained under two equal Numbers, namely (3) and (3) and so on as in the following Table.

A Table of Squares with their Genetive equal Number.

Equal Mumber	Square
1 into 1	1
2 into 2	4
3 into 3	9
4 into 4	16
5 into 5	25
6 into 6	36
7 into 7	49
8 into 8	64
9 into 9	81
10 into 10	100

Thus the Square-Number (625) is contained under two equal Numbers, viz. (25) and (25) That is to say,

[illustration]

Sectio QUADRATI (625) in quatuor Plana, à duobus Lateris (25) Segmentis, viz. (A=20) & (B=5) effecta; quorum tria ordinatim sumpta, sunt continuè proportionalia, nimirum

• 1 A quadratus=400=maximus Proportionalis
• 2 A in B=100=medius Proportionalis
• 3 B quadratus=25=minimus Proportionalis

One of the equal Numbers	25
being multiplyed by the other equal Number.	25
Makes the Product a Square Number, viz.	625

Eucl. l. 2. Theor. 4. Prop. 4. 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉, &c.

If a right Line be cut any wise into two parts, the Square made of the whole Line, is equal both to the Squares made of the Segments, and to twice a Rectangle made of the Parts.

THis holds good likewise in Numbers. For example, Let (25) be the Number of a Right Line, and that di∣vided into two parts: viz. (20) the greater, and (5) the lesser. I say,

The Square of the whole (25) is equal to the two Squares of (20) and (5) more by the two Oblongs made of (20) multiplied by (5) as in the opposite Fi∣gure may be plainly seen.

For let (20) be called	A
And (5) be called	B
A q: (or 20 multiplyed by 20) makes	400
B q: (or 5 multiplied by 5) makes	25
A, Multiplied by B, that is (20) multiplied by (5) makes	100
A, Multiplied by B, that is (20) multiplied by (5) makes	100
That is to say, these are the parts, which being united are equal to the whole Square	625

These things being premised, I say, that whereas the vulgar Rule directs (after the pointing of the Num∣ber 625 whose Square Root is to be extracted) to find out the Square-Root of the last pointed Figure on the left hand, that is (6) and not finding (6) a true Square Number, to take the next to it, viz. (4) whose Square-Root is (2.) I say, the meaning is this; That

(6) is in effect (600) which is not a true Square-Number, and the nearest to it is (400) whose Square-Root is (20.)

Again, whereas the vulgar Rule directs to subtract the Square-Number (4) out of (6) and so to find out how many times the double of the first Root is con∣tained in the Remainder and the first Figure of the next Square, viz. (22) that is 5 times, But with this provision that there may remain a Number equal to the Square of that (5) as in the example.

〈 math 〉〈 math 〉

The meaning is this, Having Subtracted A q, or the Square-Number (400) out of the Number (625) there remains (225) which is (A=20) into (B=5) or (100) for one of the Oblongs, and (A=20) into (B=5) for the other Oblong, and B q or (25) for the lesser Square. All which are the very parts of a Square, expressed in the foregoing Proposi∣tion of Euclid. Namely.

(A q) The Square of the greater Segment which is equal to—	400
(B q) The Square of the lesser Segment which is equal to—	25
(A) Into (B) one of the Rectangles—made of the Segments, and equal to	100
(A) Into (B) the other Rectangle made of the Segments and equal to—	100

The Square made of the whole (A▪••) 〈…〉〈…〉 or (25) multiplied by (25) is 〈…〉〈…〉

〈…〉〈…〉

And be the Square-Number never so great, both the manner of the Operation, and the Reason of the Ana∣lysis, or extraction of its Square-Root▪ is the very same.

But for the better understanding of all that haha been said, let the Practitioner consider w••ll the Figure Where he may evidently see, how the Root of every lesser Square being doubled, and an Unite added to it, makes up the next greater Square

Thus twice (3) or the double Root of the next less Square, more by (1) being added to that Square (9) makes up the next greater Square, viz. (16) And twice (5) or the double Root of the next less Square (25) more by (1) that is to say (11) being added to that (25) makes it (36) which is the next greater Square.