"Continued proportion is a succession of several equal ratios. Harmonical or musical proportion is a relation of several quantities, such that the first is to the last as the difference between the first two is to the difference between the last two; thus 4, 6, 12, are in harmonical proportion; for 4 is to 12 as 2 to 6.

"A root is that which multiplied into itself one or more times will produce a given number; and it is called the square root, cube root, fourth root, etc., as it has to be multiplied once, twice, three times, or more, to produce the number. Thus, 2 is the square root of 4; the cube root of 8; and the 4th root of 16. The old Pine will find it very easy to multiply roots together so as to form numbers, and somewhat difficult to find out the different roots of different numbers. A rational root is one commensurable with unity; a surd is one which is not. Thus, 2 is the rational root of 4 or of 8, or of 16. But the square root of 2, 1.4142+, is called a surd, as it cannot be exactly expressed in numbers."

"And does it not exist?" I asked.

"It does not exist in our notation, as we have no figure which when multiplied by itself will give o for the right hand figure of the result. But the old Pine must remember that arithmetic, as constructed by man, as well as every other science, is artificial, and therefore imperfect."

"In such a case," I said, "constantly as we proceed we come nearer and nearer to the correct result."

"Yes," she said, "nearer and nearer, and always nearer, but never reaching the goal. But we have seen in the case of fractions, that what can be expressed in one system of notation cannot in another. This shows the different character of sys

tems of notation, and suggests that the difficulty of expressing accurately the roots of numbers, when occurring, is a fault in the notation employed, and not that such roots are impossible. Thus, we can express the square root of 2 by a line, because of the principle in geometry that in a right angled triangle the square of the base plus the square of the perpendicular is equal to the square of the hypothenuse. But there can be no more accurate notation than the natural one of space. For this is nature's notation. And therefore it is certain that the square root of 2 can be accurately expressed."

"And," I said, "in regard to the relation of the diameter to its circumference, we have the proposition in geometry that the circumferences of circles are to each other as their radii or diameters. Hence by inversion the circumference of one circle is to its diameter as that of another circle is to its diameter. And therefore if the diameter is doubled the circumference will be doubled; or, if halved, so will the circumference be. That is, in all cases the circumference will vary as the diameter varies; which means that the ratio of the circumference to its diameter is fixed."

"Yes," she said, "we have that proposition, but its proof depends upon the theory of limits; and therefore the proposition as proposed is not demonstrated to be true. But in spite of this, Ellen thinks that the circumference of a circle and its diameter are commensurable. The only question is whether two straight lines are commensurable. For we may suppose a certain number of parallel lines to be placed contiguous to each other, so as to make a continuous surface. Inscribe a circle upon them; then must the circumference of this circle be com

posed of a certain part of each of these parallel lines, the number of lines included being determined by a second diameter at right angles to the first. This makes the circumference composed of a straight line, for instead of being composed of parts of many parallel lines it might as well be composed of different parts of the same straight line. And such straight line would represent the diameter extended."

"But," I asked, "would there not be an insurmountable difficulty in the fact that the diameter is a straight line, and the circumference curved?"

"Ellen thinks not," she answered; "for, admitting the infinite division of matter, the material forming any circumference might be rearranged in a straight line. (Ellen is assuming that any circumference and every straight line is substantial; that is, composed of a certain amount of matter, however infinitesimal the amount may be. For any existing line must be so composed. Nor does Ellen believe a non-existing line or surface is conceivable; for she doesn't believe that nothing is conceivable. And if not, no unoccupied space is conceivable. The nearest we can come to such a supposition, as Ellen thinks, is a space wherein a certain particular substance, or any particular substance, does not exist.) But a diameter is any straight line, and therefore the question whether the two are commensurable is a question whether two straight lines are commensurable, which they must be in nature's notation, because of its great adjustability, although, as in the case of the square root of 2, we can not express it in our notation.

"Ellen will not dwell upon the rules for finding the different roots of numbers, for the old Pine can find these in nearly all

arithmetics, and very plainly expounded. Besides, Ellen is going to show the old Pine all that, and lots of other interesting things, when she teaches him algebra. The square is composed of the square of tens, plus twice the product of the tens by the units, plus the square of the units. The cube is composed of the cube of the tens, plus three times the square of the tens by the units, plus three times the tens by the square of the units, plus the cube of the units. And from these principles these roots are readily deduced."

XXIII.

HE had risen, and was preparing to go. "The old Pine

SHE

has been very much interested," I said, "with Ellen's talk upon arithmetic, and looks forward with much pleasure to the future talks on mathematics which she promises. But before Ellen goes to-day he wishes she would answer a few questions upon those subjects which she has previously discussed. Ellen has said that she did not think that thought was mind."

"Then Ellen will sit down again," she said, "for the subjects are far-reaching and the old Pine is very inquisitive. Ellen doesn't think that thought is mind. She thinks mind something entirely distinct, that it makes thought; and to say that thought was mind, she thinks, would be the same as to say that a man was any of the things which he makes. For these, as Ellen thinks, bear the same relation to man as thoughts or ideas to mind.

"The old Pine will see that nothing could be more different than a man from those things which he makes-an ax, a spade or a wagon, house, or cotton factory. As far as the east is from the west, so great is the difference between the thing made and its maker; and therefore, as Ellen thinks, must God be essentially different from those things which He makes, including man, although endowing man with mind, which works, at least to a certain extent, in harmony with His own. But Ellen notices this great difference between the mind of God and