dios, will be 20; then count five of the grand Divifions, where ftop, for that is the place which reprefents 25. Where note, that if you had efteem'd the 1 at the be ginning of the Line but 1, that is, one tenth, the place which now reprefents 25, would fignify but 2. 5: Alfo if you had efteem'd it as 10, then would the place of 25, be 250; if 100, then 2500, if but or then but 025, &c. Example II. To find upon the Line the the place of 3652. First, efteem the at the beginning of the Line to be 100, then will that in the middle be 1000,and the three towards the middle 3000; from which count fix of the Grand-divifions and a half towards (4000) and then you will come to the place of 3650. Now you muft imagine the 2 to be a little beyond that half Divifion; for in this and the like examples, where we are to find 4 places, that which is Uuites must be taken by eftimation. So have you the place 3652. Note, By thefe Examples laft medtioned, you may perceive that the Figures 1, 2, 3, 4, 5, 6, 7, 8, 9, do fometimes figfy them felves alone, fometimes 10, 20, 30, &c. Sometimes 100, 200, 300, &c." As the work perform'd thereby fhall, require: The firft Figure of every Number is always that which is here fet down, and the rest ofthe Figures are to be fupplied according as the queftion fhall require. And by the varia tion and change of the power of thefe Numbers from1, to lo, or 100, or 1000, any Proportion, may be wrought by this, General Rule. Always extend the Compaffes from the firft Number to the fecond, and that diftance, or extent, applied the fame way upon the Line, fhall reach from the third to the fourth Number required. Or otherwife, extend the Compaffes from the firft Number to the third, and that extent applied the' fame way, fhall alfo reach from the fecond to the fourth. Either of thefe ways will effect the fame thing, by Examples following fhall be made appear. And it is neceffary thus to vary the Proportion, fo as to avoid the opening of the Compaffes two wide. Multiplication by the Line. as This Rule whether it be perform'd Arithmetically or Inftrumentally, depends upon Euclids Elem, Prop. 1. lib. 2. where it is demonftrated, that if two Lines be pro pos'd, whereof one is divided into diverfe parts, the Rectangle contained under those two Lines is equal to the Rectangles contained under the Line which is divided, and the parts of the Line divided. The proportion is, as one is the Multiplyer: So is the Multiplicand to the Product, Example I. Let it be required to multiply 8 by 7 the Proportion is, as I is to 8:: fo is 7: to 56. Therefore extend the Compaffes from 1 to 8; the fame extent will reach from 7 to 56, which is the product. Example II. Let it be required to Multiply 37, bys The proportion is; As 1 to 5:: fo is 37: to 185. Set one Foot of the Compaffes in 1, and extend the other Foot to 5; that fame extent will reach from 37 to 185, which is the product or 37, being Multiplied by 5. Otherwife, fet one Foot in 1 and extend the other to 37; the fame extent will reach from 5 to 15 Example |