your Compaffes, and open the Sector fo, that the Distance from 3 to 3 on the Lines of Lines be equal to that Extent; then the parallel Distance of 5 and 5, will give the Line B, fought.

In the fame manner, if B be a Line given to be diminished in the Proportion of 5 to 3, take the Line B between your Compaffes, and to it open the Sector in the Points of 5 and 5 on the Lines of Lines; then the parallel Distance between 3 and 3, will give the Line A, required.

If this way of working be not fufficient, you may multiply the Numbers given by 2, 3, 4, &c. and work by their Equimultiples; as, for 3 and 5, you may open the Sector in 6 and 10, 9 and 15, 12 and 20, 15 and 25, or in 18 or 30, &c.

PROB. III.

To divide a given Line into any Number of equal Parts:

Take the Line given between your Compaffes, and open the Sector in the Points of the Parts the given Line is to be divided into ; then keeping the Sector at this Angle, the parallel Distance between the Points of 1 and 1, will divide the given Line into the Parts required.

Fig. 9.

Example. Let AB be a Line given to be divided into 8 equal Parts. First take this Line AB between your Compaffes, and to it open the Séctor in the Points of 8 and 8; then the parallel Distance between

I and

and 1, gives the Line AC, which will divide AB into 8 equal Parts.

If the Line propofed be too long to be applied to the Legs of the Sector, divide 1 half or 1 fourth of it into the Parts propofed; and the Double or Quadruple of one of those Parts, will divide the whole Line into the Nuniber of Parts propofed.

PRO B. IV.

To find the Proportion between two or more given Liñes.

Take the greater of the given Lines between your Compaffes, and to it open the Sector in the Points of 100 and 100; then take the leffer Lines feverally, and carry them parallel to the greater, till they ftay on the fame Numbers of the Lines of Lines; and the Numbers whereon they stay, will give their Proportion to 100. Example. Let the Lines given be

AB, CD. First take the Line CD Fig. 10. between your Compaffes, and to it

open the Sector in the Points of 100 and 100; then keeping the Sector thus opened, enter the leffer Line AB parallel to the former, and you will find it cross the Lines of Lines in the Points 60 and 60; wherefore the Proportion of AB to CD, is as 60 to 100.

Or, if the Line CD be fo long that it cannot be applied to the Points 100 and 100, then you may suppose the leffer Line AB, to be 100, and cutting off CE equal to AB, you will find

D

the

the Proportion of CE to ED, to be as 100 to almoft 67; wherefore this way the Proportion of AB to CD, is as 100 to almoft 167.

This Proportion may likewife be worked by any other Numbers that admit feveral Divifions, as by the Numbers 60 and 60; and then the leffer Number will be found 36, which is, as before in leffers Numbers, as 3 to 5. This Proportion may also be worked without opening the Sector. For if the Lines between which a Proportion be fought, are laterally applied on the Lines of Lines, (or any other Scale of equal Parts) the Proportion between them will be the fame, as between the Lines to which they are equal.

PROB. V.

To find a Third Proportional to two given Lines.

the

First place both the given Lines on both Sides of the Sector from the Center, and mark the Terms of their Extenfion; than take out the fecond given Line again, and to it open Sector in the Term of the firft Line; then keeping the Sector at this Angle, the parallel Diftance between the Terms of the fecond Line, will be the third Proportional fought.

Example. Let the two Lines given Fig. 11, be AB, AC, which take out, and place on both Sides of the Sector, fo as they meet in the Center A: let the Terms of the firft Line be B, B, and the Terms of the

fecond

fecond C, C. Then take out AC, the fecond Line, again, and to it open the Sector in the Terms B, B; fo the Parallel between C, C, gives the third Proportional fought. For as AB is to AC, fo is BB equal to AC, to CC.

PRO B. VI.

Three Lines being given, to find a fourth Line proportional to them.

Place the firft and third Lines on both Sides of the Sector from the Center; then take out the fecond Line, and to it open the Sector in the Terms of the first Line. Now keeping the Sector at this Angle, the parallel Distance between the Terms of the third Line, will be the fourth Proportional fought.

Example. First take out A and C, and place them on both Sides of the Fig. 12. Sector, in AB, AC, and AD, AE, lay. ing the Beginning of both Lines in the Center A; then take out B, the fecond Line, and to it open the Sector in B, C, the Terms of the first Line; and then the Parallel between D and E, gives the fourth Proportional fought.

As in Arithmetick, it is fufficient for the first and third given Numbers to be of one Denomination, and the fecond and the fourth required to be of another, for one and the fame Denomination is not neceffarily required in them all; fo in Geometry, it is fufficient if the Sides AB, AD, refembling the first and third Lines, be measured by one Scale, and the Parallels

D 2

rallels BC, DE, be measured in another. Where fore the Proportion of A, the firft Line to C the third, being found by prop. 4. aforegoing, which fuppofe to be as 8 to 12, or in leffer Numbers, as 4 to 6, or as 2 to 3, or in greater Numbers, as 16 to 24, or 18 to 27, or 20 to 30, 30 to 45, or 40 to 60, &c. If the Sector be opened in the Points of 8 and 8, to the Quantity of B, the fecond given Line; then the Parallel between 12 and 12, will give DE, the fourth Line required. So likewife if it be opened in 4 and 4, the Parallel between 6 and 6; or if in 16 and 16, the Parallel between 24 and 24, will give the fame, DE; and fo of the rest.

PROB. VII.

Two Lines being given, to find a Mean Proportional between them.

Take the longest Line, and lay it on the Line of Lines from the Center, noting its Extenfion; then take the fhorteft Line between your Compaffes, and applying it parallelwife on the Lines of Lines, open the Sector fo, that the lateral Diftance from the Center of the Sector to the Term of this Line, be equal to the parallel Extent of the Terms of the longeft Line; and then the faid lateral Distance of the Terms of the fhortest Line, or the parallel Distance of the Terms of the longest Line, will give the Mean Proportional fought.

Note,