5. From these points, E, F, G, &c, with E1 as radius, describe the eight circles, each of which will touch two circles, and the given circle A.

To construct a foiled figure about any given regular polygon (say a hexagon.)

1. ABCDEF is the given hexagon. Bisect one side BC in G (Pr. 1).

E

B

2. With the several angles of the polygon as centres, and radius GB, describe the six arcs, which form the required hexafoil about the given hexagon ABCDEF.

SECTION IX.-PROPORTIONALS.

DEFINITIONS.

1. Ratio is the relation that one quantity bears to another of the same kind, in respect of magnitude, .e., by considering what multiple, part, or parts one is of the other. Thus, in comparing 6 with 3, we find that it has a certain magnitude with respect to 3, that is, it contains it twice; but in comparing it with 2, we find that it has a different relative magnitude, for it contains it three times.

NOTE 1.-The ratio of any two quantities (of the same kind) depends therefore on their relative, and not their absolute, magnitudes.

NOTE 2.-A "part" must be understood to mean any aliquot part (not any part).

NOTE 3.-The ratio of a to b is usually represented thusa b, or sometimes

a

2. "Proportion is the similitude of ratios" (Euc. V., Def. 8). Thus, four quantities are said to be proportionals when the first is the same multiple, part, or parts of the second that the third is of the fourth. This is usually expressed by saying a is to b as c is to d, and is thus represented, ab::c:d, or sometimes a : b = c : d.

NOTE 1. The last term (d) is called the fourth proportional. NOTE 2.-The quantities a and d are called the extremes, and b and c the means.

NOTE 3.-When four quantities are proportionals, the product of the extremes is equal to the product of the means, i.e., ad = bc.

NOTE 4.-When the two means are the same quantity-as abbc-the last term (c) is called the third proportional, and the middle term (b) is called a mean proportional.

3. A proportional in Practical Geometry is a line which bears some fixed ratio to one or more given lines. Thus, the four straight lines A, B, C, D are proportionals, D being the fourth proportional greater, and A the fourth proportional less, to the lines A, B, Č; and B, C, D, respectively

A

B

D

Again, the three straight lines A, B, C are proportionals, C being the third proportional greater, and A the third proportional less to the lines A, B; and B, C, respectively—

A
B

NOTE.-B is a mean proportional between the two lines A and C.

Problem 140.

To find a mean proportional between two given lines AB and CD.

1. Produce AB to E, and make BE equal to CD.

-D

2. Bisect AE in F (Pr. 1). With centre F, and radius FA, describe the semicircle.

3. From B, raise a perpendicular BG (Pr. 2) to meet the semicircle. Then BG is the required mean proportional between the two given lines AB and CD.

Problem 141.

To find a fourth proportional to three given straight lines, AB, CD, and EF, when the required line is less than any of the given lines.

1. Make GH equal to AB, and draw GK equal to CD, making any angle with GH.

2. Join HK, and from G, on the line GH, cut off GL equal to EF.

3. Through point L, draw LM parallel to HK (Pr. 9). Then GM is the required fourth proportional to the three given straight lines AB, CD, EF, and less than any of them.

Problem 142.

To find a fourth proportional to three given straight lines AB, CD, and EF, when the required line is greater than any of the given lines.

1. At A in AB, draw a line AG equal to CD, and at any angle with AB.

2. From A, set off AH equal to EF, and join GH.

3. Through B, draw BK parallel to GH (Pr. 9), cutting AG produced in K. Then AK is the required fourth pro

portional to the given lines AB, CD, and EF, and greater than any of them.

Problem 143.

To find a third proportional to two given straight lines AB and CD, when the required line is less than either of the given lines.

1. Make EF equal to AB, and draw EG equal to CD, making any angle with EF.

2. Join FG, and from E, with radius EG, cut EF in H.

3. Draw HK parallel to FG (Pr. 9).

Then EK is the

required third proportional to the given straight lines AB and CD, and less than either of them.