PROP. XXXII. THEOR.

If two triangles have two sides proportional, and be so placed at an angle that the homologous sides are parallel, and that the sides not homologous form the angle at which they are placed, the remaining sides form one right line.

Since the homologous sides are parallel, and the sides not homologous form an, each of the s contained by the proportional sides is to this so formed, and .. to one another; .. the given As are equiangular (Prop. 6. B. 6); therefore the threes at the point where the sides not homologous meet are together to two rights, they being to the three angles of either A; therefore, &c.

Note. The given triangles will not be capable of being so placed unless they are similar. It is evident from Prop. 4. B. 6. that any two similar As may be so placed.

PROP. XXXIII. THEOR.

In equal circles angles, whether they be at the centres or at the circumferences, have the same ratio that the arches on which they stand have to one another; so also have the

sectors.

The <s at the centre are proved to be as the arcs on which the they stand in the same way that we have demonstrated the first proposition of this book; i. e. by assuming any equisubmultiples of the antecedents, i. e. of one arc and the which stands on it, those submultiples will be contained an = number of times in their respective consequents, i. e. in the other arc and . Equisubmultiples of the antecedents are assumed by taking any submultiple of the first arc, and drawing from its extremity a radius, this radius will cut off an equisubmultiple of the first ; for the < at the centre will be divided into as many parts as the arc subtending it, if radii be drawn from the extremities of those parts into which the arc is divided. Thes at the centre being thus

proved to be as the arcs on which they stand; it follows that the s at the circumference are so.

In the proof of that part concerning the sectors, it is necessary to prove, that the sectors of the same or equal circles standing on equal arcs are ; and .. that the radius to the extremity of the assumed submultiple of the first arc cuts off an equisubmultiple of the other antecedent; this is thus proved; draw chords of arcs, these divide the sectors into triangles and segments that are respectively to one another; the As are by Prop. 4. or 8. B. 1; and the segments are Prop. 24. B. 3. for if angles be inscribed in them they shall be since they stand on arcs, i. e. on the differences between the whole

arcs and the whole circumferences;. the segments are similar, and since they stand on = bases they are = (Prop. 24. 3). Therefore the sectors are since they = are made up of As and of segments. Then whatever submultiple one of the small arcs is of the antecedent arc the same is the sector standing on this small arc of that on the antecedent; and as often as the small arc is contained in the consequent arc, so often is the small sector, &c.

Cor. 1. An angle at the centre is to four right angles, as the arc upon which it stands to the whole circumference.

For the at the centre a right :: the arc on which it stands 4th the whole circumference (Prop. 33. 6.) and it is to four rights as the arc is to the whole circumference.

Cor. 2. The arcs of unequal circles, which subtend equal angles are similar. For it is evident that they have the same ratio to the whole circumferences. (Cor. 1.) Cor. 3. It is evident that the arcs of similar segments are similar.

END OF THE SIXTH BOOK.

ADDITIONAL NOTES.

1. To cut any right line bd harmonically in a given ratio.

Fig. 2.

Draw any right line mb and make it to ds in the given ratio, then make bn bm and draw the right lines msa, ncs, then evidently ba: ad: : bc: cd.

2. Drawing a tangent at from any point in the produced diameter of a semicircle, and a perpendicular tc, the line ba is cut harmonically.

Fig. 3.

For ba: ad: : bv : do : : vt: to :: bc : cd.

3. Between the quantities ba and ad, ca is the harmonic mean, at the geometric and am the arithmetic; but the ▲ mta is a right d A, .. a t is a geometric mean between ma and ca.

4. If any point be joined with the several points of harmonic section, by the right line pb, pc, pd, på, any line b'a' intersecting them in any manner is also cut harnically.::

Fig. 4.

For through d, d' draw mdn, m'd'n' parallel to p b ; then by hypothesis ba: ad:: bc: cd, but ba: ad: : bp: dm and bc: cd: bp dn .. dm : = dn.. d'm' = d'n' .. b'p ; d'n' ::

b'p: d'm'; .. b'a' : a'd' : : b' c' : c'd'.

5. The right lines pb, pc, pd and pa are called harmonicals.

6. The consequence mentioned in Note 4, is evidently true of any right line intersecting the production of the harmonicals beyond the point of concourse p.

TO PROP. XXXIII.

7. To determine the peripheries and areas of circles.

Let it be assumed that the arc of a circle is less than the sum of the extreme tangents, and greater than the chord. Hence the periphery of the circumscribed polygon is always greater, and of the inscribed polygon less than that of the circle, however the number of sides be multiplied.

G

8. If a right line be drawn cutting the sides of a ▲ in any way, above the base and below the vertical; the sum of the sides is greater than the sum of the segments, towards the base the drawn line.

9. Hence by drawing tangents or chords to the middle points of the intercepted arcs, the number of the sides of either the inscribed or circumscribed polygons is doubled, but their perimeters are made to approach nearer to that of the circle.

10. By continuing this bisection indefinitely, the perimeters of the polygons can be made to differ from each other, and.. from the periphery of the circle by a quantity less than any assignable.

11. If regular polygons be inscribed or circumscribed to two circles, their perimeters are obviously to one another (componendo) as the radii of the circles.

12. The circumferences of circles are to each other as their radii. For let C, C be the circumferences and r,r' the radii; then if possible let r: r' : : C: C'+d; let P,P' be the perimeters of the circumscribed polygons, at the time that P' is nearer to C" than by the quantity, then C: C'+d::P:P', but P' is less than C+ Pis less than C, which is absurd. Again let if possible r: r':: C: C'd, and let p, p' be the perimeters of the inscribed polygons; at the time that p' is nearer to C' than by ; then C: C' -d::pp', but p' is greater than C'-, .. p is greater than C, which is absurd. Hencer::: C: C

13. If 1 and C2, then C-2r; the numerical value of has been found to 127 places of decimals; as a sufficiently near value however we may take ☛ = 3,14159.

14. Angles A,A' are to one another directly as their subtenses a a', and inversely as their radii, for A: 4rt. Zs:: a: C, and 4rt. Zs: A':: C': a' .. ex æquo A; A':: a: a' Ja: a'?

C: C

::

15. Similar polygons inscribed or circumscribed to two circles, have their areas to each other as the squares of the radii (19. 6.)

Such polygons can have their sides multiplied, so that their areas may differ by a quantity less than any assignable from each other, and therefore from those of the circles.

16. Hence by the same argument that has been used for the peripheries, it appears that the areas of circles are as the squares of the radii.

Pr

17. The area of the circumscribed polygon = 2

since the area of a circle can be made to differ from it by a quantity less than any assignable, the area of the circle = 1.2.

ON PORISMS.

Something still remains to be said on a very abstruse class of questions called Porisms, which are not very easily to be distinguished from Theorems and Problems.

Euclid wrote three books on them, which however have not come down to modern times. The following account of their origin and true meaning is taken from a Memoir by Playfair, in the third vol. of Edinb. Trans.

"The ancient Geometers arrived perhaps at all geometrical truths in their attempts to solve problems.

A problem was not considered solved till all the cases of it were separately discussed. This discussion soon let them see that there were circumstances in which the solution would cease to be possible, and this they perceived to happen in consequence of data becoming inconsistent.

Such instances must frequently occur in solving the simplest problems, but in the Analysis of more complex, cases occurred in which constructions failed for a directly contrary reason. Instances would be found when lines, instead of intersecting so as to afford definite solutions, or of not meeting at all so as to afford impossible ones, would actually coincide, and leave the question of course unresolved. The confusion thus arising would soon be cleared up, by observing that a problem before determined by the intersection of two lines, would now become capable of an indefinite number of solutions. This was soon perceived to arise from two parts of the requisite data becoming one; and the very curious propositions thus resulting were admitted as forming a class apart, in as much as they admitted of distinct enunciation, from Theorems and Problems, under the denomination of Porisms.

This transition of a Problem into a Porism, may be