Example III.

Let it be required to multiply 8. 75 by 3. 6. The Analogy or Proportion is, as 1. to 3. 6 So is 8. 75, to 31.5. Set one Foot in 1, and extend the other to 3. 6; the fame extent applied forward upon the Line will reach from 8.75 to 31.5.

Divifion by the Line.

In Divifion, the Quotient contains Unity as often as the Dividend does the Divifor It follows then from the 5th Defini tion to the 5th Book of Euclids Elements, that the Quotient is in Proportion to 1, as the Dividend is to the Divifor. From whence we may deduce the Proportion following, viz. As the Divifor is to Unity, fo is the Dividend to the Quotient.

Example I.

Let it be required to divide 40 by 8. The proportion is, As 8 to 1:40: 5. Set one Foot of the Compaffes in 8, and extend the other Foot to 1; that fame extent will reach from 40 downwards to 5, which is the Quotient.

I;

Otherwife extend the Compaffes from 8 to 40; the fame extent will reach from 1, to 5.

Example II.

Let it be required to divide 336 by 12. The Proportion is As 12: I:: 336: 28. Extend the Compaffes from 12 to 336; the fame extent will reach from 1 to 28; which is the Quotent required.

You know how many Figures fhould be in the Quotient, by fetting the Divifor orderly under the Dividend, &c.

The Rule of three by the Line. ·

The Rule of three by the Line must be performed by the laft General Rule.

Example I.

If 26 Acres of Land be worth 64 1. a Year; what is 36 Acres of the fame Land worth by the Year. Proportion as 26:64:: So 36 to 88. 615.

Extend the Compaffes from 26 to 64, the fame extent will reach from 36 to 88, 61 which is 88 L 12 s. 3d. 2 q. for the Answer of the Queftion.

To extract the Square Root by the Line.

Divide the space between Unity and the given Number into two equal parts:

Where

Where that Divifion falls, is the Square Root fought.

Evample II.

Ex

To find the Square Root of 36. tend the Compaffes from 1 to 36, the Middle way upon the Line between the fe two Numbers is 6, which is the Square Root of 36. In like manner you may find the Square Root of any other Number.

Of Meafurcing.

A clearer Idea of which you cannot have than that given by the Ingenious Mr Cunn, in his excellent Treatife of Fractions compleated, which is as follows.

Every Magnitude is meafur'd by fome Magnitude of the fame kind, a Line by a lineal Foot, Yard, &c. a Superficies by a fquare Foot Yard, &c. a Solid by a Cubick Foot, Yard, &c.

The Lineal Meafure is known to all

The fuperficial Meafure may be conceived, by imagining a Floor pav'd with Tiles, each a Square Foot; for then the Number of Tiles is equal to the Number of fquare Feet in that Flooring. Now if the Flooring be just one Foot broad, the Num

ber of Tiles (or of Square Feet) will be equal to the Number of Lineal Feet in the Length of the Floor; but if the Flooring be 2, 3, 4, 5, &c. Feet broad, the Num ber of Tiles, or of Square Feet, will be twice thrice, four times, five times, &c. fo many Tiles (or Square Feet.) So if the Floor were 11 Foot long and 7 Foot broad, 7 times 11 Tiles (or Square Feet) gives 77, the Number of Tiles or Square Feet in that Flooring.

The folid Meafure may be conceived by imagining a Wall built with Stones, each a Cubick Foot; then the Number of Stones will be equal to th: Number of Cubick Feet in that Wall. First therefore, if the Wall be one Foot thick and one Feet high, the Number of Stones (or Cubick Feet) will be equal to the Number of Lineal Feet in the length of that Wall. Secondly, If the Wall fhould be of the fame length and heighth one Foot as bef re, but the thickness 2, 3, 4, 5, &c. Feet (inftead of one Foot) then the Number of Stones (or Cubick Feet) will cccrdinglyhe twice,thrice,four timesfive times &c. and as many as before. Lastly, If the length and thicknefs be the fame as in the laft fuppofition, but the height (instead of one Foot) be 2, 3, 4, 5 Feet; the Number of Stones (or Cubick Feet) will be accordingly twice, thrice, four times five times, c. what it was in the foregoing. So if a

Wall

Wall is feven Foot long, three Foot thick, and one Foot high: From what has been faid, a Wall of feven Foot long, one Foot thick and one Foot high confifts of feven Cubick Feet; but a Wall of feven Foot long, three Foot thick, and one Foot high, confifts of three times feven' Cubick Feet, that is, 21 Cubick Feet. Laftly, A Wall of feven Feet long and three Foot thick as before, but five Foot high, contains five times as many, that is five times 21 Cubick Feet, or 105 Cubick Feet.

From all which is evident, that in cafting up any Menfuration, the Multiplier in any of the Multiplications is an Abflract number as well as in all other Multiplications whatfoever, which may prevent the falle Confequences ufually drawn from multiplying Feet by Feet, viz. That of multiplying by a contract number,as 31, 19s. od:by 31:19 5:or Half aCrown by half aCrown; which is contrary to the Nature of Multiplication, whofe Operations are only compendious additions, either of the Multiplicand or fome part of it, continually to its felf or its part.

Of Tyling:

Suppofe a Roof in length 120 Foot on both Sides, and the depth of one Side 18 Foot; thefe two numbers multiplied together produce 2160 Foot, which is 21 Squares, and