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, 10 31.5. Set one Foot in 1, and extend the other to 3. 6; the same extent applied forward upon

the Line will reach from 8.75 to 31.5.

Division by the Line. In Divifion, the Quotient contains Unity as often as the Dividend does the Divi. for : It follows then from the 5th Defini. tion to the sth Book of Euclids Elements, that the Quotient is in Proportion to 1, as the Dividend is to the Divisor. From whence we may deduce the Proportion following, viz. As tbe Divifor is to Unity, lo is the Dividend to the Quotient.

Example I. Let it be required to divide 40 by 8. The proportion is, As 8: to ! ::40:5. Set one foot of the Coinpaffes in 8, and extend the other Foot to i; that same extent will reach from 40 downwards to 5, which is the Quotient.

Otherwise extend the Compasses from 8 to 40; the same extent will reach from 1, tos.

Example II.


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

Youl know how many Figures should be in the Quotient, by ferring the Divisor orderly under the Dividend, &c.

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

Example I.

If 25 Acres of Land be worth 64 l. 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 Compasses from 26 to 64, the faine extent will reach from 36 to 88, 61 which is 88 l. 12 s. 3d. 2 q. for the Answer of the Question.

To extra& the Square Root by the Line.

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


Where that Division falls, is the Square Root fought.

Evample IT.
To find the Square Root of 36.

Ertend the Coinpasses from 1 to 36, the Middle way upon thc Line between these two Numbers is 6, which is the Square Root of 36. In like manner yoi may find the Square Root of any other Number.

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

Every Magnitude is meafur'd by some Magnitude of the fame kind, a Line by a lineal Foot, Yard, 6. a Superficies by a {quare Foot Yard, &c. a Solid by a Cubick Foot, "Yard, & c.

The Lineal Moafurc is known to all.

The superficial Measure 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 square Feet in that Floorirg. Now if the Flooring be just one Foot broad, the Nuinber of Tiles (or of Square Feet) will be equal to the Number of Lineal Feet in the Length of the Floor ; but it the Flooring be 2, 3, 4, 5, Sc. Feet broad, the Number of Tiles, or of Square Feet, will be twice thrice, four times, five times, & many Tiles (or Square Feet.) So if the Floor were u 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 Measure 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, it 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 should be of the same length and heighth one Foot as bef re, but the thickness 2, 3, 4, 5, &c. Feet (instead of one Foot); then the Number of Stones (or Cubick Feet) will cccrdinglyhe twice, thrice, four timesfivé. times &c. and as inany as before. Lastly,If the length and thickness be the same as in the last suppofition, but the height (instead of one Foot) be 2, 3, 4, 5 Feet; the Number

, of Stones (or Çubick Feet) will be accordingly twice, thrice, four times five times, &c. what it was in the foregoing. So if a


Wall is seven Fout long, three Foot thick, and one Foot high: From what has been said, a Wall of seven Foot long, one Foot thick and one Foot high consists of seven Cubick Feet ; but a Wall of seven Foot long, three Foot thick, and one Foot high, consists of three times feven Cubick Feet, that is, 21 Cubick Feet. Lafly, A Wall of seven Feet long and three Foot thick as before, but five Foot high, contains five times as inany, that is five times 21 Cubick Feet, or 105 Cubick Feet.

From all which is evident, that in cafting up any Mensuration, the Multiplier in any of the Multiplications is an Abflract number as well as in all oiher Multiplications whatsoever, which may prevent the false Confe. quences usually drawn from multiplying Feet by Feet, viz. That of inultiplying by a contract number as 31. 195. od:by 31:19 3:or Half aCrown by halt a Crown; which is contrary to the Nature of Maltiplication, whose Operations are only compendious additions, either of the Multiplicand or fome part

of it, continually to its self or its part.

Of Tyling: Suppose a Roof in length 120 Foot on both Sides, and the depth of one side 18 Foot; these two numbers multiplied together produce 2160 Foot, which is 21 Squares, and

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