the figures it is that constitute the period of the first importance, then ought we to call this portion of figures, whether it consist of one, two, or three figures; then ought we to call it the FIRST PERIOD.

273. Besides the circumstances thus enumerated, by which this first period of figures is exalted, there is another circumstance by which it is yet more emphatically distinguished; this first period will always contain that important portion of the number, called the primary cube.

274. With this preparation we may proceed to the consideration of the method of extracting the cube root of any given number; or, in other words, as we have before seen, of the method of finding what would be the dimension of any cube, formed of a given number of other cubes of a given dimension.

275. Let it be, then, that we have to find, what would be ths size of a cube formed of 79507 cubic inches.

276. We know, in the first instance, that the root of this number, consisting of two periods, will be two figures; and we further know, that as there are in it but five figures, the first, or highest period, will be left with only two figures, that is, with 79... which, however, is 79000.

277. Considering the matter for a moment, or turning to the Table of Primary Cube Numbers, we find that the primary cube in this case is 64, having 4 for its root.

278. However, it is not 64 units, but 64 thousands; nor is the root 4, but 40; for the 4 is to have another figure after it, which will raise it to the place of tens.

279. Thus have we, on the outset, as our primary cube, a cube formed of 64000 inch cubes, and being, in length, or breadth, or height, 40 inches.

280. So far all is simple enough, and we have, if we may use the terms, consumed, or employed 64,000 of the 79 thousand odd inch cubes, leaving a remainder of 15507 inch cubes yet to be disposed of.

281. Now the question comes, how are these 15507 inch cubes to be disposed of; and in what manner are they to be applied to the augmentation of the primary cube?

282. To repeat the operation just passed, that is, to extract the cube root of this remaining quantity, will not tend towards our purpose, for it would give us another cube, and another remainder, on which remainder, if we were to repeat the operation, we should find another, and another cube, each smaller than the former. It is not to a series of cubes, declining in size, that we would reduce the 79507 inch cubes, but to a single cube, which should comprise this number of inch cubes.

283. We have employed 64000 of the given number of cubes, and have, as before stated, 15507, that is, nearly one fourth of the number remaining: so the question continues to present itself, how shall this remainder be disposed of in augmentation, as it must be, of the primary cube?

284. Had we, in substance, the primary cube of 40 inches actually before us, and these 15507 inch cubes, there are two modes of employing them in its augmentation; we might, as one of the modes, first form a sort of bed, or stratum, of the inch cubes, a square bed, the size of the primary cube, that is, 40 inches square, and, placing this cube upon it, we

might pile up, as a sort of wall around the four sides, a quantity of these remaining small cubes; and finally, in order to make all the sides equal, place a layer of the small cubes, similar to that below, upon the upper side of the primary cube, thus enveloping it equally on all sides, and so preserving its cubical form. This, I say, had we the materials before us, is one of the modes in which we might proceed to accomplish our object, and, in that accomplishment, to dispose of the given materials.

285. The other mode-and there are but twoof attaining both these ends, is, instead of placing a layer or stratum of the small cubes upon each of the six sides of the primary cube, to place these layers upon three of the sides only; for instance, upon the top, on one side, and then, on what may be called one end. This method accomplishes the purpose effectually as the other, and is more easy, and more simple.

286. "More easy, and more simple:" These ought to determine our choice between the two modes; so let us proceed upon the latter, the more simple and more easy mode.

287. We have a cube of 40

inches, and a parcel of inch cubes 08 90 to be added to it. We have determined that the best method of making the addition to the first or primary cube, is to place the small cubes in strata on three of its sides, in such manner, of course, as will preserve its cubical form.

288. The primary cube is 40 inches. A layer for one of its sides, that is to say, 40 inches square, will require 1600 of the inch cubes. Three of these layers;

that is, one for each of the three sides, will require 4800 of the inch cubes. But we have 15507 of them. That is more than three times-but not four timesthe number requisite to form three complete layers. Three times the number required to form three single layers, would make these layers each three inches thick, and these would consume, or require, 14400 of our inch cubes. Let us, then, place three layers, each being of these dimensions, upon the appropriate sides of the primary cube; and, then, how will the matter stand; the figure will not be a cube, but will be of the form represented in the margin; that is, a cube covered on three of its sides, with strata or layers, of its own

dimension, and sufficient, therefore, to cover, each its proper side of the cube; but, extending no further, the ends of the layers remain uncovered, and there is, consequently, a deficiency; the cube being incomplete.

289. Directing our attention to this deficiency, we soon perceive that three long pieces, or prisms, each the length of the primary cube, that is, 40 inches, and of the thickness of the layers, that is, 3 inches; we soon perceive that three of these would go far to supply the deficiencies.-Three inches is the thickness of the layers. The pieces required to supply the deficiencies must, therefore, be each three inches square, and 40 inches long. Three inches square, that is 3 x 3 is 9, which multiplied by the length, which is 40, make 360 inches; and this is the quantity required to form one of these prisms. But we want three of them. Three times 360 are 1080. For which we have a sufficiency in our remainder.

290. And do these complete the cube? Do these

three pieces, of three inches square and forty inches long, supply the entire deficiencies which we observed in the last figure? They do

not. Added to the layers which were before placed on the primary cube, they make those. layers-which were before 40 inches-43 inches square, save and except the small deficiency, a sort of knotch, observable in the figure in the

margin, which deficiency is occasioned by the want of length in some one of the three prisms, to cover the ends of the two others, and thereby to complete the cube. The ends of these prisms are each three inches square: a cube of three inches would, therefore, cover them, and thus is the cube completed: a cube of 43 inches.

291. Let us now review the work we have done. And, first, as to the quantity of the small cubes employed in the construction of this cube which we have just completed.

292. Turning back to paragraph 275, and tracing the matter through the process, we see, that in the construction of this cube, there has been employed, of the inch cubes,

293. And this is the number, the cube root of which we proposed to extract. This is the number of inch cubes which we proposed to erect into a single cube. Or, rather, we proposed to inquire, what would be the dimension of a cube constructed of 79507 inch