boxes for holding a pencil or tracing point are brought to the

in the same straight line, and which is divided at P in the determinate ratio. While the point C, therefore, is carried along the boundaries of any figure, the intermediate point P will, by the scholium, trace out a similar figure, reduced in the proportion of OC to OP or of OB to OD, and which, in the present instance, is that of three to one.

But the point P may be placed in the fulcrum, the tracer inserted at O, and the crayon held at C; in which case, C would delineate a figure which is enlarged in the ratio of OP to PC or of OD to DB. If the points O and P were now brought to coincide with A and E, the distances AE and EC being equal, the original figure would be transferred into a copy exactly of the same dimensions.

In reducing small figures, however, artists commonly prefer another method, which is partly mechanical. The original is divided into a number of small squares, by means of equidistant and intersecting parallels. Other reduced squares are drawn for the copy, which is then filled up, by observing the same relative position and form of the boundaries.-One material advantage results from this practice; for if oblongs be used in the copy instead of squares, the original figure will be more reduced in one dimension than another, which is often very convenient where height and distance are represented on different scales.

8. Proposition twenty-eight. The curious properties of the crescents, or lunulæ, contained in the first corollary, were discovered by Hippocrates of Chios, in his attempts to square the circle. But a beautiful extension of them was briefly suggested by the Reverend Mr Lawson, and afterwards explained and demonstrated by Dr Hutton of Woolwich, in whose ingenious Mathematical Tracts it now appears. It is a mode of dividing a given circle into equal portions, and contained within equal circular boundaries. For example, let it be required to cut the circle APBQ into five equal spaces. Divide the diameter AB into five equal parts at the points C, D, E and F; on AC, AD, AE, and AF describe the semicircles AGC, AID, ALE, and ANF, and on BC, BD, BE, and BF, towards the op

P

N

L

B

D

E SFO

M

K

H

Q

posite side, describe the semicircles BHC, BKD, BME, and BOF; the circle APBQ will be divided into five equal portions, by the equal compound semicircumferences AGCHB, AIDKB, ALEMB, and ANFOB.

For the diameter AB is to the diameter AD, as the circumference of AB to the circumference of AD, or (V. 3.), as the semicircumference APB to the semicircumference AID; and AB is to BD, as the semicircumference APB to the semicircumference BKD. Wherefore (V. 20.) AB is to AD and BD together as the semicircumference APB to the compound boundary AIDKB; and consequently these interior boundaries AGCHB, AIDKB, ALEMB, and ANFOB, are all equal to the semicircumference of the original circle.

Again, the circle on AB is to the circles on AE and AF, as the square of AB to the squares of AE and AF; and consequently (V. 20.) the circle on AB is to the difference between the circles on AE and AF, as the square of AB to the difference between the squares of AE and AF, that is (II. 17.), the rectangle under the sum and difference of AE and AF, or

twice the rectangle under EF and AS, the distance of A from the middle point of EF. Whence the circle APBQ is to the difference of the semicircles ALE and ANF, or the space ALEFN, as the square of AB to the rectangle under AS and EF; and, for the same reason, the circle APBQ is to the space FOBME, as the square of AB is to the rectangle under BS and EF; consequently (V. 20.) the circle APBQ is to the compound space ALEMBOFN, as the square of AB to the rectangles under AS and EF and BS and EF, or the rectangle under AB and EF; but the square of AB is to the rectangle under AB and EF, (V. 25. cor. 2.) as AB to EF, which is the fifth part of AB; wherefore (V. 5.) any of the intermediate spaces, such as ALEMBOFN, is the fifth part of the whole circle.

B

9. Proposition twenty-ninth. This elegant theorem admits of an algebraical investigation. Put AC-a, AB=b, BC=c, and let s denote the semiperimeter, and T the area of the triangle; then, by Prop. 23. Book II., 2AC.CD=a2+c2-b2, consequently CD= 2+c2_b2

2a

‚a2

and BD2=BC2—CD2—

D C

a2+c2-b2), and, therefore, by Prop. 5. Book II., T2=

2a

But this expression, consisting of the difference of two squares, may be decomposed, by Prop. 17. Book II.; whence T2= 2ac+a2+c2—b2 2ac—a2—c2+b2 _ (a+c)2—b2 b2—(a—c)2

4

4

and, decomposing these factors again,

4

T2_a+b+c a−b+c a+b―c ¬a+b+c

= s, a−b+c

4

2
a+b+c

2

2

2

Now,

-b,

2

2

sa; wherefore we obtain, by substitution,

T= √(s(sa) (s—b) (s—c)).

Suppose the sides of the triangle to be 13, 14, and 15; then

the area is (21.8.7.6) ≈ 7056 84. If the sides were 21, 17 and 10, the area would be the same, for (24.3.7.14)= 705684.

This most useful proposition was known to the Arabians, but seems to have been re-invented in Europe about the latter part of the fifteenth century.

Another corollary might be subjoined to this proposition : As the semiperimeter of a triangle is to its excess above the base, so is the rectangle under its excesses above the two sides to the square of the radius of the inscribed circle.

For BI: BG:: EI: DG, and consequently (V. 25. cor. 2.)BI: BG:: EI.DG: DG; but it was proved that EI.DG is equivalent to AG.AI, and hence BI : BG :: AG.AI: DG2. Now BI has been shown to be the semiperimeter, and BG, AG and AI its excesses above the base and the other two sides of the triangle, of which DG is the radius

of the inscribed circle.

Hence let the sides of the triangle be denoted by a, b and c, and the semiperimeter by S; the square of the radius of the S-a. S-b. S-c. inscribed circle will then be expressed by

S

Suppose, for example, the sides of the triangle were 13, 14 and 15, the radius of the inscribed circle would be the square root of or of 16, that is 4.

8.7.6
21

Employing the same notation, it is not difficult to perceive that the continued product of all the sides of a triangle must be equivalent to the product of twice their sum into the radii of the inscribed and circumscribing circles. Thus, 13.14.15273084.4.81.

Recurring to the last figure, it is evident that BG: BI :: DG: EI : DG.EI : : EI, or, since DG.EI AG.AI, BG: BI:: AG.AI: EI; that is, As the excess of the perimeter above the base is to the semiperimeter itself, so is the rectangle under its excesses above the other two sides of the triangle to the square of the radius of the circle of external contact below the base. Thus, in the triangle taken for illustration, 6:21::8.7:196, and consequently the radius of the circle under the base is 14. Again, 7:21:: 8.6: 144, and the radius of the circle touching externally the side 14 is therefore 12. And, in the same manner, 8:21:: 7.6: 1104; which gives 10 for the radius of the circle applied beyond the shortest side 13.

10. Proposition thirtieth. A similar and very important problem, which formerly occupied a place in the text, must not be omitted. It likewise furnishes an ingenious and concise approximation to the quadrature of the circle, first published at Padua in the year 1668, by James Gregory, my illustrious predecessor in the mathematical chair of the University of Edinburgh; and seems the more deserving of attention, as it probably led that original author to the investigation of the Method of Series.

Given the area of an inscribed, and that of a circumscribed, regular polygon; to find the areas of inscribed and circumscribed regular polygons, having double the number of sides.

Let TKNQ and HBDF be given similar inscribed and circumscribed rectilineal figures; it is required thence to determine the surfaces of the corresponding inscribed and circumscribed polygons AKCNEQGT and VILMOPRS, which have twice the number of sides.

From the centre of the circle, draw radiating lines to all the angular points. It is evident that the triangles ZXK and ZAB are like portions of the given inscribed and circumscribed figures TKNQ and HBDF; and that the triangle ZAK, and the quadrilateral figure ZAIK are also like portions of the derivative polygons AKCNEQGT and VILMOPRS. And since