from the centre F shall not fall entirely within the circle described from the centre G; the condition that A and C are greater than B, ensures that one of these circles shall not fall entirely without the other. Hence the circles must meet. It is easy to see this as Simson says, but there is something arbitrary in Euclid's selection of what is to be demonstrated and what is to be seen, and Simson's language suggests that he was really conscious of this.

I. 24. In the construction, the condition that DE is to be the side which is not greater than the other, was added by Simson; unless this condition be added there will be three cases to consider, for F may fall on EG, or above EG, or below EG. It may be objected that even if Simson's condition be added, it ought to be shewn that F will fall below EG. Simson accordingly says "...it is very easy to perceive, that DG being equal to DF, the point G is in the circumference of a circle described from the centre D at the distance DF, and must be in that part of it which is above the straight line EF, because DG falls above DF, the angle EDG being greater than the angle EDF." Or we may shew it in the following manner. Let H denote the point of intersection of DF and EG. Then, the angle DHG is greater than the angle DEG, by I. 16; the angle DEG is not less than the angle DGE, by I. 19; therefore the angle DHG is greater than the angle DGH. Therefore DH is less than DG, by I. 20. Therefore DH is less than DF.

If Simson's condition be omitted, we shall have two other cases to consider besides that in Euclid. If F falls on EG, it is obvious that EF is less than EG. If F falls above EG, the sum of DF and EF is less than the sum of DG and EG, by I. 21; and therefore EF is less than EG.

I. 26. It will appear after I. 32 that two triangles which have two angles of the one equal to two angles of the other, each to each, have also their third angles equal. Hence we are able to include the two cases of I. 26 in one enunciation thus; if two triangles have all the angles of the one respectively equal to all the angles of the other, each to each, and have also a side of the one, opposite to any angle, equal to the side opposite to the equal angle in the other, the triangles shall be equal in all respects.

The first twenty-six propositions constitute a distinct section

of the first Book of the Elements. The principal results are those contained in Propositions 4, 8, and 26; in each of these Propositions it is shewn that two triangles which agree in three respects agree entirely. There are two other cases which will naturally occur to a student to consider besides those in Euclid; namely, (1) when two triangles have the three angles of the one respectively equal to the three angles of the other, (2) when two triangles have two sides of the one equal to two sides of the other, each to each, and an angle opposite to one side of one triangle equal to the angle opposite to the equal side of the other triangle. In the first of these two cases the student will easily see, after reading I. 29, that the two triangles are not necessarily equal. In the second case also the triangles are not necessarily equal, as may be shewn by an example; in the figure of I. 11, suppose the straight line FB drawn; then in the two triangles FBE, FBD, the side FB and the angle FBC are common, and the side FE is equal to the side FD, but the triangles are not equal in all respects. In certain cases, however, the triangles will be equal in all respects, as will be seen from a proposition which we shall now demonstrate.

If two triangles have two sides of the one equal to two sides of the other, each to each, and the angles opposite to a pair of equal sides equal; then if the angles opposite to the other pair of equal sides be both acute, or both obtuse, or if one of them be a right angle, the two triangles are equal in all respects.

If the angle B be equal to the angle E, the triangles A BC, DEF are equal in all respects, by I. 4. If the angle B be not equal to the angle E, one of them must be greater than the other; suppose the angle B greater than the angle E, and make the angle ABG equal to the angle E. Then the triangles ABG, DEF are equal in all respects, by I. 26; therefore BG is equal to EF, and the angle BGA is equal to the angle EFD. But the angle EFD is acute, by hypothesis; therefore the angle BGA is acute. Therefore the angle BGC is obtuse, by I. 13. But it has

been shewn that BG is equal to EF; and EF is equal to BC, by hypothesis; therefore BG is equal to BC. Therefore the angle BGC is equal to the angle BCG, by I. 5; and the angle BCG is acute, by hypothesis; therefore the angle BGC is acute. But BGC was shewn to be ob

tuse; which is absurd. Therefore the angles ABC, DEF are not unequal; that is, they are equal. Therefore the triangles A BC, DEF are equal in all respects, by I. 4.

Next, suppose the angles at C and F obtuse angles.

The demonstration is similar to the above.

Lastly, suppose one of the angles a right angle, namely, the angle C. If the angle B be not equal to the angle E, make the

A

E

angle ABG equal to the angle E. Then it may be shewn, as before, that BG is equal to BC, and therefore the angle BGC is equal to the angle BCG, that is, equal to a right angle. Therefore two angles of the triangle BGC are equal to two right angles; which is impossible, by I. 17. Therefore the angles ABC and DEF are not unequal; that is, they are equal. Therefore the triangles ABC, DEF are equal in all respects, by I. 4.

If the angles A and D are both right angles, or both obtuse, the angles C and F must be both acute, by I. 17. If AB is less than BC, and DE less than EF, the angles at C and F must be both acute, by I. 18 and I. 17.

The propositions from I. 28 to I. 34 inclusive may be said to constitute the second section of the first Book of the Elements. They relate to the theory of parallel straight lines. In I. 29 Euclid uses for the first time his twelfth axiom. The theory of parallel straight lines has always been considered the great difficulty of elementary geometry, and many attempts have been made

to overcome this difficulty in a better way than Euclid has done. We shall not give an account of these attempts. The student who wishes to examine them may consult Camerer's Euclid, Gergonne's Annales de Mathématiques, Volumes XV and XVI, the work by Colonel Perronet Thompson entitled Geometry without Axioms, the article Parallels in the English Cyclopædia, a memoir by Professor Baden Powell in the second volume of the Memoirs of the Ashmolean Society, an article by M. Bouniakofsky in the Bulletin de l'Académie Impériale, Volume v, St Pétersbourg, 1863, articles in the volumes of the Philosophical Magazine for 1856 and 1857, and a dissertation entitled Sur un point de l'histoire de la Géométrie chez les Grecs...... par A. J. H. Vincent. Paris, 1857.

Speaking generally it may be said that the methods which differ substantially from Euclid's involve, in the first place an axiom as difficult as his, and then an intricate series of propositions; while in Euclid's method after the axiom is once admitted the remaining process is simple and clear.

One modification of Euclid's axiom has been proposed, which appears to diminish the difficulty of the subject. This consists in assuming instead of Euclid's axiom the following; two intersecting straight lines cannot be both parallel to a third straight line. The propositions in the Elements are then demonstrated as in Euclid up to I. 28, inclusive. Then, in I. 29, we proceed with Euclid up to the words, "therefore the angles BGH, GHD are less than two right angles." We then infer that BGH and GHD must meet: because if a straight line be drawn through G so as to make the interior angles together equal to two right angles this straight line will be parallel to CD, by I. 28; and, by our axiom, there cannot be two parallels to CD, both passing through G.

This form of making the necessary assumption has been recommended by various eminent mathematicians, among whom may be mentioned Playfair and De Morgan. By postponing the consideration of the axiom until it is wanted, that is, until after I. 28, and then presenting it in the form here given, the theory of parallel straight lines appears to be treated in the easiest manner that has hitherto been proposed.

I. 30. Here we may in the same way shew that if AB and EF are each of them parallel to CD, they are parallel to each other. It has been said that the case considered in the text is so obvious as to need no demonstration; for if AB and CD can

never meet EF, which lies between them, they cannot meet one

another.

I. 32. The corollaries to I. 32 were added by Simson. In the second corollary it ought to be stated what is meant by an exterior angle of a rectilineal figure. At each angular point let one of the sides meeting at that point be produced; then the exterior angle at that point is the angle contained between this produced part and the side which is not produced. Either of the sides inay be produced, for the two angles which can thus be obtained are equal, by I. 15.

The rectilineal figures to which Euclid confines himself are those in which the angles all face inwards; we may here however notice another class of figures. In the accompanying diagram the angle AFC faces outwards, and it is an angle less than two right angles; this angle however is not one of the interior

E

angles of the figure AEDCF. We may consider the corre sponding interior angle to be the excess of four right angles above the angle AFC; such an angle, greater than two right angles, is called a re-entrant angle.

The first of the corollaries to I. 32 is true for a figure which has a re-entrant angle or re-entrant angles; but the second is not.

I. 32. If two triangles have two angles of the one equal to two angles of the other each to each they shall also have their third angles equal. This is a very important result, which is often required in the Elements. The student should notice how this result is established on Euclid's principles. By Axioms II and 2 one pair of right angles is equal to any other pair of right angles. Then, by I. 32, the three angles of one triangle are together equal to the three angles of any other triangle. Then, by Axiom 2, the sum of the two angles of one triangle is equal to the sum of the two equal angles of the other; and then, by Axiom 3, the third angles are equal.

After I. 32 we can draw a straight line at right angles to a given straight line from its extremity, without producing the given straight line.

Let AB be the given straight line. It is required to draw from A a straight line at right angles to AB.