these, the problems, should be carefully constructed by the student in the manner prescribed by Euclid, with the exception of the inscription of regular figures in a circle in the fourth book, which may be better done by the table of chords already mentioned, or by trial, which is a very good exercise in the use of the compasses. Of the second class we have said sufficient: of the third class, the principal propositions are Book I. 35, 43, 47; Book II. 12, 31; Book III. 35, 36. Our limits will hardly allow us to enter at full length into the ocular demonstration of these propositions; we will nevertheless shortly indicate some methods which will be found useful.

Book I. Prop. 32. If the three angles of a triangle be cut off and applied round the same point, the first and last segments of the sides will lie in the same straight line or let ABC be a triangle, of which the greater angle is C (if the triangle be acute-angled, any angle may be chosen ;) draw CD perpendicular to AB, and EF through E and F the middle points of CA and CB. Cut out the triangle, and double CEF over the line EF, so that C and D coincide: if BD and AD be then doubled, so that B and A meet in D, it will be found that the three angles of the triangle are so arranged as to show their equality with two right angles. The demonstrations of the corollaries of this proposition, as given in Euclid, are, or may be made, ocular.

Book I. Prop. 35, 43. The areas which are here called equal cannot be made to coincide; but it is evident in the demonstrations, that the same areas added to both form figures which can be made to coincide.

Book I. Prop. 47, may be demonstrated to the eye

in the following way: let ABC be the right-angled triangle, of which the right angle is at B. On AC describe the square AGHC (the letters go round the square) turned, so that the triangle ABC may fall within it. On AB, the larger side, take AD equal to BC the smaller side, and join DG: on GD take GE equal to BC, and join EH: produce CB to meet EH in F. Cut out the triangles ABC, ADG, GEH, and HFC, and give them with the remaining square EFBD to the learner as a dissected puzzle, which he is to put together into one square, and also into two squares side by side. He will soon find this out, and will see that the single square is that on the hypothenuse, while the two are those on the sides. To verify this proposition in numbers, when he has learnt how to estimate the area of a square, it will be desirable to have a method of finding right-angled triangles whose sides are whole numbers. The following is the most simple: take any two numbers (4 and 7), multiply each by itself (16 and 49); take the sum and difference of the last (65 and 33): these are the hypothenuse and one of the sides of a right-angled triangle: the other side is twice the product of the numbers first chosen (56.)

Book II. 12, 13. The simplest method of verifying these propositions is by means of numbers, and the following rule will find a triangle whose three sides, together with the perpendicular let fall on one side from the opposite angle, and the segments made by that perpendicular, are all whole numbers. Let ABC be the triangle, CD the perpendicular from C on AB, making the segments AD, DB. Choose any even number for the perpendicular CD, which has two even divisors; 8 for example, which is divisible by 2 and 4. Multiply

and divide it by one of the even divisors, 2 for example, giving 16 and 4. Take the half sum and half difference of these, giving 10 and 6. If the first be CB, the second is DB. Proceed in the same way with the second divisor 4, which gives 17 and 15. If the first be AC, the second is AD. The side AB is the sum or

difference of AD and DB, that is, 21 or 9, according as we construct CDA and CDB on the same or opposite sides of CD; that is, according as we choose both the angles A and B acute, or one of them obtuse.

When propositions have been verified in the case of whole numbers, verification in some other case should be attempted by measurement and calculation. This will not be so satisfactory, on account of the errors which must arise in estimating fractions of the parts of the scale. Nevertheless, it should be attempted, and the source of error pointed out. It will be found that such numerical application will fix the proposition in the mind.

We shall now proceed to show how the reasoning of geometry may be taught, on the supposition hat the student is perfectly master of the language and facts of at least a large portion of the first four books of Euclid. Every beginner will not be competent to do as much as we have pointed out, but many will; and we are convinced that, by so doing, they will be prepared to look upon the reasoning of Euclid, as what it really is, one of the most admirable results of human thought, and not, as is often the case, to regard it as most unprofitable drudgery. If Euclid be usually ill-understood at first, it is because, with the usual quantity of preparation, one reading of the first six books is no more than is necessary to learn the terms and modes

of speech, and to acquire an indefinite notion that there is something to be learnt, and a sort of aperçu of the leading facts. These, we submit, ought to be known beforehand, if the reasoning part is to be made a source of enjoyment. We receive with more satisfaction the information that charcoal and the diamond, two well-known substances imagined to be distinct, are the same, than we should have done, had we only to regard the latter as a new product, arising from the crystallization of the former.

As we have already observed, the more important part of the study of geometry is the habit of reasoning which should be acquired from a science where all is absolute demonstration. If we should be thought to speak rather of what ought to be than of what is, it must be owing to something in the method of teaching which keeps the logic of geometry in the background, and substitutes for it an exercise of less improving character, which may be the mere learning by rote the phraseology of propositions. When we find, as it frequently happens, that by A is greater than B, and B is greater than C, much more then is A greater than C,' students imagine it to be proved that A is much greater than C, we think it evident that their attention has not been directed, in the least degree, to the nature of the connexion which exists between one part of a syllogism and another.

If we were to propose to ten students who, as times go, have read the first four books of Euclid, the following argument,- Every equilateral triangle is equiangular, and an equiangular triangle may contain less than an acre, therefore an equilateral triangle may contain less than an acre'-we are convinced that nine out of ten would admit the conclusion without scruple, and

that of those nine not two would find out where the fallacy lay, even when the instructor assured them that there was a fallacy. The three propositions are evident truths: the falsehood implied in the little word 'therefore' passes unheeded, because that same word has never been anything more than a mere pleonasm in their previous course of reasoning. Finding the demonstrations employed threw no great light upon the matter, they preferred believing the propositions upon the evidence of their senses, which is the way by which most young persons gain the assurance which they have of the truth of the facts asserted. The reasoning was said over, that is, the angles and sides of triangles were repeated in endless confusion, mixed with a decent distribution of the words because,' by hypothesis,' and therefore.' To show that this is not a notion of our own, a mere march-of-mind complaint against existing institutions, we will again cite the editor whom we quoted in the former part of this article:- Once upon a time a certain father, resolving not to be imposed upon by reports, determined to examine into his son's progress in this science, produced the book, and required him to demonstrate a proposition to which he referred: the young man, though unacquainted with the subject, taking courage from his father's ignorance, began very impudently in some such manner as follows:-Because the angle ABC is equal to the angle CBA, therefore the angle DEF is equal to the angle CEF, &c., ringing the changes upon sides and angles, until he had spun out his demonstration to a decent length, and then kept silence in expectation of his father's opinion, who, with a grave and important countenance, remarked, "This is what we call demonstration."