in the demonstration we have the angles ABC and ABE equal to two right angles, and also the angles ABC and ABD equal to two right angles; and then the former two right angles are equal to the latter two right angles by the aid of the eleventh axiom. Many modern editions of Euclid however refer only to the first axiom, as if that alone were sufficient; a similar remark applies to the demonstrations of Í. 15, and I. 28. In these cases we have omitted the reference purposely, in order to avoid perplexing a beginner; but when his attention is thus drawn to the circumstance he will see that the first and eleventh axioms are both used.

We may observe that errors, in the references with respect to the eleventh axiom, occur in other places in many modern edi. tions of Euclid. Thus for example in III. 1, at the step "therefore the angle FDB is equal to the angle GDB," a reference is given to the first axiom instead of to the eleventh.

There seems no objection on Euclid's principles to the following demonstration of his eleventh axiom.

Let AB be at right angles to DAC at the point A, and EF at right angles to HEG at the point E: then shall the angles BAC and FEG be equal.

Take any length AC, and make AD, EH, EG all equal to AC. Now apply HEG to DAC, so that H may be on D, and HG on DC, and B and F on the same side of DC; then G will coincide with C, and E with A. Also EF shall coincide with AB; for if not, suppose, if possible, that it takes a different position as AK. Then the angle DAK is equal to the angle HEF, and the angle CAK to the angle GEF; but the angles HEF and GEF are equal, by hypothesis; therefore the angles DAK and CAK are equal. But the angles DAB and CAB are also equal, by hypothesis; and the angle CAB is greater than the angle CAK; there

fore the angle DAB is greater than the angle CAK. Much more then is the angle DAK greater than the angle CAK. But the angle DAK was shewn to be equal to the angle CAK; which is absurd. Therefore EF must coincide with AB; and therefore the angle FEG coincides with the angle BAC, and is equal to it.

I. 18, I. 19. In order to assist the student in remembering which of these two propositions is demonstrated directly and which indirectly, it may be observed that the order is similar to that in I. 5 and I. 6.

I. 20. "Proclus, in his commentary, relates, that the Epicureans derided Prop. 20, as being manifest even to asses, and needing no demonstration; and his answer is, that though the truth of it be manifest to our senses, yet it is science which must give the reason why two sides of a triangle are greater than the third: but the right answer to this objection against this and the 21st, and some other plain propositions, is, that the number of axioms ought not to be increased without necessity, as it must be if these propositions be not demonstrated." Simson.

I. 21. Here it must be carefully observed that the two straight lines are to be drawn from the ends of the side of the triangle. If this condition be omitted the two straight lines will not necessarily be less than two sides of the triangle.

I. 22. "Some authors blame Euclid because he does not demonstrate that the two circles made use of in the construction of this problem must cut one another: but this is very plain from the determination he has given, namely, that any two of the straight lines DF, FG, GH, must be greater than the third. For who is so dull, though only beginning to learn the Elements, as not to perceive that the circle described from the centre F, at the distance FD, must meet FH betwixt F and H, because FD is less than FH; and that for the like reason, the circle described from the centre G, at the distance GH...must meet DG betwixt D and G; and that these circles must meet one another, because FD and GH are together greater than FG?" Simson.

The condition that B and C are greater than A, ensures that the circle described from the centre G shall not fall entirely within the circle described from the centre F; the condition that A and B are greater than C, ensures that the circle described

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. 18; therefore the angle DHG is greater than the angle DGH. Therefore DH is less than DG, by I. 19. 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 an-
gle 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
not unequal; that is, they are equal.
ABC, DEF are equal in all respects, by I. 4.

angles ABC, DEF are Therefore the triangles

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

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. 27 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