times with something over. This having been tried. sufficiently, let the lines ac and A C be placed one upon the other, a coinciding with A, and let the same thing be done with ab and A B, and let them be carried on by the compasses as in the adjoining diagram, the larger figures above being placed at the end of multiples. of the line denoted by large letters, and the smaller figures below denoting the same with respect to the small lettered line. This construction should be made as correctly as possible, and the successive points should be made by a small circle, cutting the line, and not by forcing the compasses into the paper.

It will be observed, on comparing the larger and smaller figures in both lines, that they run in the same order with respect to one another: for example, we find six times ac a little greater than twice AC; we also find six times ab a little greater than twice AB. Again, three times AB falls between eight times and nine times ab; and three times AC also falls between eight times and nine times ac. Such arithmetical conclusions as the following should also be brought forward. As twice AC is nearly six times ac, ac is nearly two-sixths of AC; similarly ab is nearly two-sixths of AB. Cases may be taken in which AB and ab have a simple common measure, in which cases one of the greater figures will soon fall on the same point as one of the lesser in both lines. The constant similarity of the way in which the greater figures are distributed among the lesser may be

made the ground-work of the definition of proportion : care being taken to show that whenever ac is any arithmetical fraction nearly of AC, ab is nearly the same fraction of AB. The preceding method may be introduced as a way of determining nearly what fraction a line is of another line. We must, however, leave the instructor to his own judgment, as to how far his pupil can bear anything beyond the first and simplest step.

ence.

With regard to our second preliminary, the measure of an angle, the proportionality of the circumference of a circle to its radius will be much more easily explained as a fact, than it will be afterwards deduced as an inferThe division of the circle into degrees needs no observation, except that, unless the learner be taught very clearly how to distinguish between the angle of one degree, and the arc of one degree, he will be liable to some confusion, which will not be mended by his finding it asserted in the books of analysis which he subsequently reads, that when the radius is unity, the arc of one degree, and the angle of one degree, are the same things. Instead of a protractor, which is at best but a clumsy instrument, he had better be furnished with a table of chords to every degree, and taught how to use it, taking the radius (which should be 100) off a common scale. It would also be well that he should form this table for himself for every ten degrees, to the same radius, dividing the circle by trial several times, and taking the mean of several of his determinations. This would be a practical illustration of a refined and useful process, and it would tend to fix notions of the value of repeated observations in his mind, when he came to compare his own single and average determinations with a better table. We need hardly instance the use

of the circle in giving accurate conceptions of angles greater than two right angles.

The way of finding the area of a rectangle is, like the doctrine of proportion, perfectly simple, when the sides of the figure can be expressed in whole numbers. There is no need to dwell on the way of showing that a rectangle whose adjacent sides are three and four inches respectively contains 3×4, or 12 square inches. There is some difficulty in passing to the area of the rectangle whose sides are 3 and 43 square inches. It must first be made evident, that the addition of any fraction of itself to one side only, adds the same fraction of the rectangle to the rectangle, which can be done, in simple instances, by construction. For example, the rectangle whose sides are 3 and 4 inches having the side 3 increased to 3, that is, having one-twelfth of itself added: the first rectangle, by the division of the side 3 inches into 12 parts, can be divided into 12 such strips as that strip by the addition of which the new rectangle is made. But the first rectangle is 12 square inches, therefore the second is 13 square inches; each of the strips being, though not in form, yet in magnitude, one square inch. This second rectangle being obtained. suppose the side 4 inches to increase to 4 inches, or to have two parts out of 20, or one part out of 10, added to itself. By a process similar to the preceding it may be shown, that the second rectangle, 13 square inches, has also one part out of ten added to itself, or 1 square inches, giving for the whole 14 square inches. The sides being 3 and 4, or 3 and 4 inches, the arithmetical multiplication of these gives or 143, or 14, the number of square inches just found: verifying the principle which the foregoing investigation is intended.

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to fix in the mind of the student, that the arithmetical rule of multiplication applied to the units or fractions of units in the sides, gives the square units, or fractions of square units, which are in the surface of the rectangle.

The preceding principles contain difficulties which it is needless to disguise; but when it is considered that the doctrine of proportion, the measurement of angles, and of the areas of rectangles, are the principal foundations of the application of arithmetic to geometry, it will appear that the trouble of explaining them clearly is not wasted. The difficulty is in the subject itself; the learner has not been accustomed to such considerations, and it is therefore hopeless to expect that he can avoid all perplexity. It would be easy to confine ourselves to the most simple view of the case: to consider nothing but equimultiples in proportion, or rectangles with whole numbers of units in their sides. But the time gained here would be more than lost when the student comes to the more complex forms of his propositions; and a judicious teacher will cast his eye beyond the present moment, and, though he may defer matters of minor importance, will not shrink from those of greater difficulty and importance when he knows that any neglect will oblige him to slur over future propositions, and leave them not only half finished, but with the idea in the mind of the learner that they are quite complete. To take a very common method of proceeding, let us suppose it has been explained how to find the area of a rectangle whose sides are expressed in whole numbers, and the rule Multiply the sides, or the units in the sides, together' has been obtained. The instructor neglects either to establish the rule for fractions, or even to mention the incompleteness of the pre

ceding case, and the matter passes by. In some future lesson a rectangle occurs, whose sides are fractional, and there is a stoppage. The instructor says, 'You know what you did before; repeat the rule,' and the learner accordingly repeats it, and is made to multiply the fractions in the sides: having previously learned a rule which he has been taught to call multiplication of fractions. Here is an evident fallacy, being the assumption without proof of the following proposition :-If the answer to a question which contains whole numbers be correctly deduced from the multiplication of those numbers, then the same question, when fractions are substituted, is correctly solved by multiplying the numerators of the fractions for a numerator, and their denominator for a denominator. Though we have thrown aside for the present the reasoning which connects one proposition with another, we do not therefore say that it would be wise to introduce fallacies, or to encourage unwarranted assumptions, by making them preliminary to sound reasoning.

The whole of the propositions in the first four books which we have not marked as immaterial, may be divided into three classes: the first containing the problems; the second the theorems in which some equality is asserted which may be verified by cutting out some parts of the figure and laying them over others, such as that any angles in the same segment of a circle are equal. The third class contains those theorems in which areas are asserted to be equal to other areas, differing in form from themselves, though not in magnitude, such as Book I. 47, in which it is asserted that the sum of the squares on the sides of a right-angled triangle, is equal to the square on the hypothenuse. The first class of