several steps of this addition, we are really adding up wrong

figures; and, consequently, that, as far as the influence of these reaches, our final product must be erroneous. I will give you an example. Suppose we have to multiply 27-14986 by 92-41035; and suppose, moreover, that these decimals had resulted from curtailing others of greater extent; as, for instance, 27·149855213, &c., and 92·41034604, &c. we should obtain the product of our proposed factors, as in the margin: but if, instead of five places of decimals in our factors, we had taken six, we should have had

27.14986

92.41035

1|3574930

8144958

271 4986

10859 944

54299 72 2443487 4

2508 928 065051

27.149855 × 92-410346 2508.927|49439983.

It is very plain, therefore, that the small error introduced into the fifth decimal of the factors, employed in the margin, has sufficient influence on the product, to render all the decimals of it, after the first three, widely erroneous. Even the third decimal differs by a unit from the more correct product above; but this latter is itself not strictly accurate, because advanced decimals have still been omitted in the factors: if the 213, &c., had been included in our multiplicand, all the decimals, beyond the vertical line, that is, beyond the 7, would have been affected; so that the 4, at present next to the 7, would have been a 5: we may conclude, therefore, that the product found in the margin is true, to the nearest unit, as far as three places of decimals; but that all the figures beyond these three should be expunged, as necessarily erroneous. In most books of Arithmetic, you are told, that these advanced decimals should be omitted, because they are superfluous, giving to the result a degree of minute accuracy not usually requisite in practical matters: but you see, from what is here shown, that they should be omitted, because no confidence can be placed in them, because, in fact, they are all wrong, and are no more worthy of being retained in our result than any row of figures written down at random in their place.

(85.) The practical conclusion you are to draw from what has now been said is this: when you multiply two factors together, the decimals in which have been curtailed, as here supposed, in adding up the partial products, disregard the

G

sums of all the columns up to that column, inclusive, in which the final figure of the last partial product is placed, and retain only the decimals furnished by the remaining columns. The last of the decimals thus retained should be increased by unit, if the first of the dismissed figures be a 5, or a greater number. In the marginal work above, a vertical line is drawn, cutting off the columns, of which the sums contribute nothing but inaccuracy to the result. It will, of course, occur to you, that it would save much waste labour if we could be spared the work of these inaccurate columns; and you will be glad to find that this may be done by a very simple contrivance. It was easy to foresee, before commencing the operation above, that seven decimals would have to be suppressed in the result; and, therefore, that three decimals only were to be retained: our object, then, would be to limit our operation to just so much work as would be necessary to furnish us with these three decimals; but as it is .desirable that we should know what the fourth decimal would be, in order that the third may be as correct as we can make it; that is, in order that the third may be increased by a unit, should the fourth be a high figure, we ought to be able to get four decimals in our result, and then to limit it to three, which may be presumed to be correct in the last figure to the nearest unit: we have, therefore, to multiply 27.14986 by 92-41035, so as to give only four places of decimals in the product: this is done as follows: place the units figure of the multiplier under the fourth decimal of the multiplicand, and then write all the other figures of the multiplier, so as that the entire row may be reversed: then, in proceeding with this inverted multiplier, observe the following caution: reject all the figures of the multiplicand which lie to the right of the figure by which you multiply, carrying, however, from these rejected figures, whatever would have been carried if they had been retained; and write the first figure you get, in each partial product, in the same vertical line, as in the margin, and you will thus find the product, 2508.928, true to three places of decimals. The figure in the units place of the given multiplier being 2, this 2 is first put under the fourth decimal figure of the multiplicand; after which the inverted multiplier is completed, and the work

27.14986

5301429

24434874

542997 108599

2715

81

14

2508.9280

carried on agreeably to the preceding directions. By comparing it with the more lengthy operation before given, you will see that the partial products, as far as they are required, arise in reverse order, and are correct, as far as they go, to the nearest unit: you will, of course, observe, that, in the carryings from the rejected figures of the multiplicand, the uniform principle of compensating for a rejected 5, or greater figure, by adding a unit to the figure on the left of it, is to be attended to, and applied: thus, in multiplying by the 1, the product, arising from the 9 in the multiplicand, on the right, is rejected, but a compensating 1 is carried to the next product; that is, we say, once 4 is 4, and 1 carried makes 5. In like manner, when we reach the last figure, 5, we say, 5 times 7 are 35; carry 4: 5 times 2 are 10, and 4 are 14. This example, with the explanations that have accompanied it, will sufficiently prepare you for the following rule.

(86.) To find the Product of Two Factors, containing Decimals, to a proposed Number of Places.

RULE 1. Count, from the decimal point in the multiplicand, as many decimals, annexing zeros if the decimals are too few, as you wish to secure decimal places in the product.

2. Under the last of these, put the unit-figure of the multiplier, or a zero, if there be no unit-figure, and then introduce all the other figures of it, so that the entire multiplier may appear with its figures in reverse order.

3. Multiply by the several figures of this reversed multiplier, neglecting, however, all those in the multiplicand to the right of the figure you are using, but, at the same time, carrying what would be carried, if nothing were neglected, and, moreover, carrying an additional unit, if 5 or a greater figure be rejected from the product.

4. Let each terminating figure of the partial products thus found, be in one vertical column; the first column to be summed up in the addition process: then, when this process is completed, you will have the product required, the decimal point being so placed, as to mark off the proposed number of decimals.

(87.) When the decimals in each of your factors are strictly true to the last figure, and your product is to be applied to a purpose, for which so many exact decimals as would make up the number in both the factors are not necessary, you may, by this rule, limit the number brought out to as few as you

please. It is, therefore, matter of choice with you, whether, in such a case as this, you take the trouble to work your example in full, and thus give to your result a needless degree of minute accuracy, or content yourself with only the amount of accuracy really wanted; and use, for this end, the contracted method: but remember, you have no choice when the decimals in your factors are not thus each of them complete and accurate in the final figure; you must then use the contracted method, not to dispense with needless accuracy, as above, but in order to preclude absolute error. In this case, you should count all the figures of that factor which contains the greater number, and so many figures of the uncontracted product, cut off from the right hand, should be expunged, not as merely useless, but as erroneous. You must, therefore, so apply the preceding rule, as to exclude, from the product, just this number of figures. NOTE, If one of the factors be quite correct, then only so many figures as this contains are to be rejected.

The following examples will sufficiently illustrate the application of the rule.

1. Multiply 348-8414 by 51.30742, so as to preserve only four decimals in the product.

Here, reversing the multiplier, after having taken care to put the 1 in the multiplier, under the fourth decimal of the multiplicand, we see, that a vacant place occurs in the multiplicand over the final figure 5 of the reversed multiplier we therefore supply this vacant place, by putting in it a 0, and then multiply, as in the margin. The result may be considered as correct, as far as it goes, provided the factors producing it, have no error in their last decimals; but, if we are not assured of this, then we cannot depend upon more than the first two decimals, for since the complete product would have nine decimals, and that each factor has seven figures, 9-7= 2, expresses the greatest number of decimal places in the product that can be relied upon, with any confidence. The work would, therefore, be as here annexed: the 1 in the multiplier being now placed under the second decimal of the multiplicand. As the 2 in the multiplier has no figure above it, the product by this 2 is, of course, 0, but as the

348.84140

2470315

174420700

3488414 1046524

24419

1395

70

17898.1522

348.8414 24703150

1744207

34884 10465

244

14

1

17898-15

product of the preceding 3 by the 2 is so great as 6, we carry 1 on that account, and the insertion of this 1 completes the multiplications.

2. Find the product of 339377 and 325, to as many places of decimals as can be depended upon.

•339377

5230

101813

6787

1697

Here, as the multiplier, which is without error, has three figures, and as there are six decimals in the factors, only three decimals are to be preserved in the product, which is, therefore, 110-297. If we had computed to four places of decimals, we should have got 110-2975. As already noticed, we may always compute to one place more than the number of places to be preserved, and may increase the last of the preserved figures by unit, if the additional figure be so great as 5: in the present case, 110.297 and 110.298, may be considered to be about equally correct.

Exercises.

110-297

1. Multiply 480-14936 by 2.72416, and retain in the product only four decimals.

2. Multiply 15-917127 by 30.31667, retaining as many decimals as may safely be depended upon.

3. Multiply 1.7958563 by 30.31667, to four places of decimals.

4. Multiply 62311052 by 170, to six places of decimals, which is one more than can be strictly depended upon. 5. Multiply 1.628894 by 214.87, retaining no decimals that cannot be relied on.

6. Multiply 81.4632 by 7.24651, retaining only three decimals.

7. Multiply 3.7719214 by 4471618, retaining all the decimals to be depended upon, namely, six.

8. Multiply 053407 by 047126, retaining all the decimals that are likely to be correct.

9. Multiply 325-701428 by 7218393, preserving only three decimals in the product.

10. Multiply 63942, &c. by 53217, &c.

NOTE. It is proper to state here, that from our ignorance of the true value of the decimals suppressed in our factors, and compensated for by a modification of the last decimal that is retained, and from the like modification of the last decimal in certain of our partial products, we cannot always be quite sure that the last decimal in our contracted product is invariably true to the nearest unit. It may in unfavourable cases err to the