CONTENTS. THEIR NATURE, MODE OF STATEMENT, &c. THEIR ORIGIN, REDUCTION, ADDITION, SUBTRACTION, MULTIPLICA- TION, AND DIVISION. THEIR NATURE, Uses, and mode of Statement. THEIR ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. REDUCTION OF VULGAR FRACTIONS TO DECIMALS. OF DUODECIMALS; TWELFTHS, OR CROSS MULTIPLICATION: EXTRACTION OF BiQuadrate, &c. ROOTS. 4 OF PROGRESSION AND PROPORTION, AND OF THE RATIOS OF NUMBERS. 1. PROPORTION and RATIOS are, perhaps, the most beautiful and interesting, as they likewise constitute the most useful branch of the science of numbers; most beautiful for the harmony and the order which they exhibit, and most useful in the frequency of their application to affairs of business and of civil life. With regard to progression, it is a very simple matter; a sort of notation, merely, of proportional numbers. 2. Progression is advancement; and the word is applied to the manner in which any series of numbers advance or increase by any sort of regular and orderly progress; as do 2, 6, 10, 14, 18; or 5, 7, 9, 11, 13; or 1, 2, 3, 4, 5; each of which lines, or series, of figures increases, as you see, at each step, by the repeated addition, in the first, of four, in the second, of two, and in the third, of one. And the word progression is, also, applied, although not quite so properly, to series of numbers which, in a mode equally regular, decrease as they proceed, as do 18, 14, 10, 6, 2. And, whilst the former may be called increasing series, those of the latter description are called decreasing series. 3. But there is another mode of progression, a mode in which the increase is produced, not by the repeated addition of one number, but by a repeated multiplication by one number, as in the following series, 2, 8, 32, 128, 512, and so on; the number with which we multiply, in this instance, being four. 4. Now these two modes of progression have, as different things ought to have, different names. That mode which is produced by Addition, is called Arithmetical Progression; whilst that which is produced by Multiplication, is called Geometrical Progression. The reasons for adopting these two names are not very obvious; and to inquire into those reasons would lead us out of our course. But it is necessary thus to distinguish the two modes. With regard to the numbers, or, as they are called, TERMS, forming both the modes of progression, they have several very curious, and some of them very valuable properties; on some of which properties we may with advantage observe. 5. In the first place, with regard to any series of numbers increasing, or decreasing, in arithmetical progression, thus, 4, 6, 8, or 8, 6, 4, it is to be observed, that the first and the last number in each series, added together, are equal to twice the middle number. And the reason of this, as you cannot fail to see, is, that as the numbers increase, or decrease, by even, or equal steps, so the number on one side. is as much more, as that on the other side is less, than the middle number; so that, put the two together, and they make twice the middle number: and this is true of every such series, whether the steps by which numbers increase or decrease be great or small. 6. But this, which is true with regard to series such as the above, consisting of three numbers, is likewise true of any other regular series of numbers or terms, however extensive the series, and whether the increase, or the decrease, at each step, as I have just stated, be great or small. For, let us suppose a series of terms consisting of a thousand; or rather, in order that we may have a middle term, let the series consist of a thousand and one. Is it not evident, that as there will be five hundred steps of increase on one hand of the middle term, and five hundred of decrease on the other, every step being equal; is it not evident, that what is lost on one hand is gained on the other; and that if we add the last term on each hand together, we shall have, as in the above short series of three terms, just twice the middle term? And, on this principle it is, that in measuring the trunks of trees, as timber, the measurement as to thickness, is taken by girthing them around the middle, that is, at an equal distance from each end. And this is regarded as the true measurement, or as it is called, the average thickness; seeing that, whatever the trunk may loose by tapering towards one end, is gained by its increase towards the other. 7. Again, that which is true of the two extreme terms of a series of numbers, whether that series be long or short, is true of any other two terms, taken one on either hand, an equal number of steps from the middle term. And this is true of all such pairs of terms, for the reason above stated; namely, that as much being gained on the one hand, at every step, as is lost by every step on the other, any two terms taken at an equal number of steps from the middle term, will be equal to twice the middle term; and equal too, for the same reason, to any two other terms taken in like manner, however near, or however distant from the middle term. 8. Yet, again: take a line of four terms; thus, 5, 6, 7, 8. Now here is not one, but two middle terms; and the extreme terms being each equally distant from the middle terms, whatever one extreme falls short of the two middle terms added together, is made up by the other extreme; so that, the two extremes are equal to the two means; for, by this name," means," are these middle terms called. And this, which is true with regard to this series, is, for the same reason, true with regard to those of any other series; and true, also, with regard to any other two terms of any other series, such terms being taken thus at eqnal distance from the two middle terms. 9. A consequence of the relationship thus existing between a series of numbers of this description is this; that if we be informed of three, of almost any three particulars respecting such series, we can tell all the other particulars respecting it. As, for example, if we have the first and the last terms, and the sum of the whole, we can then tell the number of terms, and the rate of increase at each step; or, as it is called, the rate of progression. Again, having the first term, the rate of progression, and the number of terms, we can tell the last term, and the sum of all the terms. But here is quite sufficient for our present purpose, in arithmetical progression. Now, therefore, for the other, and more important kind of progression; that is, when the increase is produced by a repeated multiplication by one number, which, as before stated, is called geometrical progression; a mode of progression, which produces numbers bearing towards each other a relationship very different from that which is produced by the successive addition of the same number to each successive term. And, for the purpose of more distinctly marking this dif ference, let us here set down a series of terms in each of the modes of progression. And, further, in order that the difference may be most distinctly seen, let us, in each series begin with the same number, and let the increase be made in each by the use of the number three: thus, 5, 8, 11, 14, 17, 20, 23, Arithmetical Progr. 5, 15, 45 135, 405, 1215, 3645, Geometrical Progr. 10. Now, the vast increase which the latter series makes, compared with the former, is not the point on which I have to remark, but, in the first place, the difference in the relationship which the several |