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(7.) In 115200 grains apothecaries' weight, how many scruples, drams, ounces, and pounds ?
Ans. 5760 sc. 1920 dr. 240 oz. 20 lbs. (8.) In 432 nails cloth measure, how many yards ?
Ans. 27 yds. (9.) In 18240 poles how many miles ? Ans. 57 mls. (10.) How many years of 365 days each in 10512000 minutes ?
Ans. 20 yrs. (11.) How many acres, roods, and perches are there in 4459 perches ?
Ans. 27 a. 3 r. 19 p. (12.) How many acres in 87120 yards. ? Ans. 18 a.
To prove any sum in this rule : if in Reduction Descending, reverse it, and do it by Reduction Ascending. Reduction Ascending is proved by Reduction Descending.
THE RULE OF THREE. In all our former lessons, in the knowledge acquired by the pupil from the sums he has heretofore worked and the rules he has practised, we have had but one aim and end in view ; that was, to lead him to this rule, and by it to teach him to turn that knowledge to account, to apply to advantage the rules heretofore studied, which taught the pupil to join numbers together and to separate them. This rule teaches that that joining and that separating may be applied to enable us to carry on the various trades, and to perform with ease and credit the various calculations that civilization imposes, or that our station in life demands.
To obtain a perfect knowledge of a rule so important, too much pains and too much research cannot be employed by the learner, and to enable him to study with benefit to himself the few examples I will insert for practice, it will be necessary for him to read this short lesson upon progression and proportion, and the ratios o numbers; which teaches the principle on which all rules having the adjustment of proportions for their object are conducted.
PROGRESSION AND PROPORTION, Progression is gradual and regular enlargement or increase; and the word is applied in this instance to any series of numbers which increases by a regular progress, as 2, 4, 6, or 4, 8, 12, 16, 20; each of which series of figures increases by repeated addition; in the first instance, of two at each step ; and in the second, of four.
And we also find the word progression applied, but not with equal propriety, to another series of numbers, which, in a mode equally regular, decrease, as 6, 4, 2, or 20, 16, 12, 8, 4; now while we may call the former an increasing series, we call the latter a decreasing series.
Besides this mode there is another, and in this other, instead of repeated additions of one number, we have repeated multiplications; taking the first of the above series, it will be 2, 4, 8, and the second 4, 16, 64, 256, 1024; the numbers by which we multiply in the first instance being 2, and in the second 4.
Now, reverting again to the former series, and comparing the second line in that, where we increased by repeated additions, with the second in that where the increase is made by continued multiplications, and to shew clearly the difference of the result, we will write them one under the other, thus : 4, 8, 12, 16,
64, 256, 1024, 2nd.
In the first, after proceeding five steps, adding four at each, our fifth term is 20 ; in the second, after proceeding five steps, and multiplying by four at each step, our fifth is 1024. Now two modes of progression, differing so much in their result, require two names to distinguish them. We accordingly call the first Arithmetical Progression, and the second Geometrical Progression.
With regard to the first of our series, Arithmetical Progression, taking the first three numbers, 2, 4, 6, we find that the first and third numbers, if added, are double the centre number; and if we take the second series, 4, 8, 12, 16, 20, and add the first and last numbers together, we find that they are also double as much as the centre number; and what is true of these two series, is true also of any other series whose proportion is preserved in like manner, even were the series to ex. tend to any number of places, for taking both extremes at equal distance from the centre, as much as the numbers increase on one side, they decrease on the other.; hence it is that, in measuring the trunks of trees, we girth them round the centre to ascertain the average thickness; for whatever the trunk may lose by tapering towards one end, is gained by its increase towards the other. If we add another term to the above ; thus, 4, 8,
24, and increase its number of places to six, we then have two middle terms instead of one; the extremes are equally distant, and if added together, the amount is equal to that of the two middle terms. Again, what is true of the two extremes, is true of any other two terms taken at equal distance on either hand from the two centre terms, or, as they are called, means.
The consequence of this relationship existing between a series of numbers is, that if we have the extremes and
the sum of the whole, we can tell the number of terms, and the rate of progression.
But enough of Arithmetical Progression. Let us now study that which is more essential for our present purpose, I mean Geometrical Progression.
I have before pointed out the difference in increase between the two modes of progression.
Let us now insert the same line again, not to point out the difference of increase, but to examine the relation that one term bears towards another.
4, 16, 64, 256, 1024. Now any series and every series having the same properties, it will be sufficient to describe the properties of one, to enable the pupil thoroughly to understand them. In the above we have five terms, each of which is a factor of that which comes after it, and a product of those before it; so that however long the series may be, there is the same proportion between the first and second term, as there is between the last term and the one on its left; one term being 3, 6, or some certain number of times greater or less than any of the others. 16 is four times as many as 4, and 4 is four times less than 16.
This mode of progression has the same properties that the other mode we were speaking of has, namely, having any three terms, as the first, last, and middle term or terms, we can find the rest.
But our present purpose, and the reason I have at all troubled the pupil with it, is, to shew how the Rule of Three is grounded upon this very progression.
We see in the above series, that 4 multiplied by 4 increases to 16, and 256 multiplied by 4 increases to 1024, and 4 bears the same proportion to 16 that 256 does to 1024; that is to say, 16 is four times as much as 4, and 1024 is four times as much as 256.
Another property to be observed in this series is, that if you take the two extremes, 4 and 1024, and multiply them together, their product, 4096, will be equalled by the product of the middle term, 64, multiplied by itself, 4096; and also, that if you multiply the two next equidistant terms, the same result will follow; and further we observe, that in a series where the terms are even, the two centre terms multiplied will produce the same sum as the two extremes, or as any other two terms in the series taken equi-distant from them.
Again we find, that any one term bears the same proportion to another, however distant, as any other two terms, taken at an equal distance, bear towards one another.
Taking the same series, 4, 16, 64, 256, 1024, four bears the same proportion to 256, as 16 does to 1024; it would be in any series, however many terms there may be in it, and however distant the terms selected may be from one another.
To distinguish these couplets, we have terms by which we express them ; for instance, 4 and 256; we call the 4 antecedent, or before going, and 256 the consequent. You have also observed that 256 is 64 times as much as 4, and also that 1024 is 64 times as much as 16; this 64 is called the rate or ratio of the proportion subsisting between these terms; and however large or small the ratio may be between terms, yet in all cases the properties I have described exist.
We have hitherto considered these proportionals as a continued series; that is to say, beginning with four, and increasing regularly in a fourfold proportion, or by a ratio of four. Nothing of this sort is required to form proportional numbers. In the order they were consi.