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In the laft Example, the given Number is .3758; which confifting of four Places, two Cyphers must be annexed, to make fix Places, as it is a Decimal,

If the Learner has a Mind, he may prove the Examples, by multiplying the Roots by themselves, and that Product by the Root, as was done in the firft Example: Where there is a Remainder, as in the fifth, feventh, eighth, and eleventh Examples, it must be added to the laft Product: Thus, in Example 11,

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I fhall only add one Example more, where a Cypher arises in the Root.

Exam. 12. To extract the Cube Root of 29.508381,

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2508381 Refolvend.

270000

900

270900 Divifor,

81

8100

270000

278181

9

2503629 Subtrahend.

4752 Remainder.

The Divifor 2790 cannot be had in 2508 the Refolvend; therefore place a Cypher in the Root, and take down 2508; to which joining 381, the next Period, it gives 2508381, a new Refolvend; and then proceed in the Work as before.

Art. 51. To
To multiply Feet, Inches, and
Parts, by Feet, Inches, and Parts.

TH

HE Multiplication of Feet, Inches, and Parts, commonly called Cross-Multiplication, being very useful, I fhall endeavour to make it as eafy to the Learner as I can, by a new Method, the best I have yet feen; for which I am obliged to Mr. Thomas Triplett, of Kingston, who was fo kind as to communicate it to me.

When Feet are multiplied by Feet, the Product is Feet; if Feet are multiplied by Inches, or Inches by Feet, the Product is Inches; but, if Inches are multiplied by Inches, the Product will be Parts, of which twelve make an Inch: From whence it arifes, that Learners are often at a Lofs where to fet down their Product: To prevent which Difficulty, Mr. Triplett invented the following Method: Divide the Multiplicand by 12, and, that being multiplied by the Inches, each Product is et under the Denomination from whence it is produced.

Exam. 1.

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Exam. 1. The Multiplicand and Multiplier being fet down, draw a Line over the Multiplicand; then divide the Multiplicand by 12, thus, the twelfth Part of 9 Feet is 9 Inches; fet 9 over the Inches in the former Multiplicand: Allo the twelfth Part of 6 Inches is 6 Parts; which muft, therefore, be placed one Place to the right Hand, after making Dots betwixt, as is ufual in fuch Cafes, and as is done above.

Having thus made two Multiplicands, (the new one for the Inches of the Multiplier, and the other for the Feet) multiply the 3 Inches into the 6 Parts, and it makes 18, which, being divided by 12, gives 1, and 6 remains; fet down the 6 Parts under the 6 Parts in the upper Multiplicand, and carry the 1; then, 3 times 9 is 27, and I that was carried is 28, which, being divided by 12, gives 2, and 4 remains; fet down the 4, in the Place of Inches, under the 9, and the 2 in the Place of Feet: Then multiply the 9 Feet and 6 Inches by the 4 Feet in the Multiplier, faying, 4 times 6 is 24, which, being Inches, must be divided by 12; and 12 may be had twice in 24, and nothing remains; therefore, fet down o in the Place of Inches, and carry 2; laftly, 4. times 9 is 36, and 2 that was carried is 38, which, being Feet by Feet, must be fet down entire, in the Place of Feet.

Under these two Products draw a Line, and add them together; by which it appears, that the Product of 9 Feet and 6 Inches, by 4 Feet and 3 Inches, is 40 Feet, 4 Inches, and 6 Parts, or Half an Inch,

Exam. 2. The 12 Feet in the Multiplicand, being divided by 12, gives 1; which fet down in the Place of Feet, over the former, having first drawn a Line; and, as there is nothing remaining, put a Cypher in the Place of Inches: Then the 7

Inches, being divided by 12, gives 7 Parts, which must be "placed to the right Hand.

Then multiply the 9 Inches in the Multiplier into the I Foot and 7 Parts, the upper Multiplicand, thus, 9 times 7 is 63, which, being divided by 12, gives 5, and 3 remains; fet down the 3 in the Place of Parts, under the 7, from which it was produced, and carry the 5 to the next; which being a Cypher, 9 times nothing is nothing, but the 5 that was carried is 5, which fet down in the Place of Inches; then, 9 times I is 9, which put down in the Place of Feet: Next multiply the 12 Feet and 7 Inches, the lower Multiplicand, by the 6 Feet in the Multiplier, thus, 6 Feet by 7 Inches is 42 Inches, or 3 Feet and 6 Inches; put down the 6 Inches under the 5 Inches, and carry the 3; then, 6 times 12 is 72, and the 3 that was carried is 75, which put under the 9 Feet; now add thefe two Products, and their Sum, 84 Feet, 11 Inches, and 3 Parts, is the true Anfwer.

Before I proceed to any more Examples, it may be proper to observe, that, an Inch being the twelfth Part of a Foot, and what we call a Part the twelfth Part of an Inch, there can be no Number fo large as 12, except in the Feet: Therefore, in dividing the Multiplicand by 12, nothing more is required, the Numbers being under 12, than to remove them a Place lower; for the twelfth Part of any Number of Feet, under 12, makes the fame Number of Inches; and the twelfth Part of the Inches, they always being under 12, are so many Parts. The Multiplicand being thus reduced to one Twelfth of what it was, every Product will ftand under the Denomination from whence it is produced, when multiplied by the Inches in the Multiplier: Otherwife, multiplying by Inches reduces cach Kind a Place lower, which is here already done.

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Exam. 3. Draw a Line over the Multiplicand, which, being divided by 12, gives 1 Foot, 3 Inches, and 5 Parts; for 12 may be had once in 15, and 3 over; put down the I over the Feet, and the 3 over the Place of Inches, and the 5 Inches in the old Multiplicand put in the Place of Parts in the new Multiplicand: Multiply the 1 Foot, 3 Inches, and 5 Parts, by 10, thus, 10 times 5 makes 50, which, being divided by 12, gives 4, and 2 remains; fet down the 2 under the 5, and carry 4; then, 10 times 3 is 30, and 4 that was carried is 34, which, being divided by 12, gives 2, and 10 remains ; fet down the 10 Inches, and carry 2 to the Feet; saying, 10 times I is 10, and 2 that was carried makes 12, which, being Feet, must be fet down entire: Then multiply the 15 Feet and 5 Inches by the 9 Feet, thus, 9 Feet by 5 Inches is 45 Inches, or 3 Feet and 9 Inches; put down the 9 Inches under the 10 Inches, and carry the 3; then, 9_Feet by 5 Feet is 45 Feet, and 3 that was carried is 48 Feet; put down the 8 Feet under the 2, Feet, and carry the 4; laftly, 9 times I is 9, and 4 that was carried is 13, which put down as in the Example: Now draw a Line under these two Products, and add them; the 2 Parts put down; but the 10 Inches, being added to the 9 Inches, makes 19 Inches, or 1 Foot and 7 Inches; put down the 7 Inches, and carry the I Foot to the Feet, which are added in the ufual Manner.

Exam. 4. Draw a Line over the Multiplicand, and the 17 Inches in the Multiplicand, being divided by 12, gives 1 Foot and 5 Inches; place the 1 over the Feet, and the 5 Inches place over the Inches, and the 8 Inches in the old Multiplicand put in the Place of Parts in the new Multiplicand, for 8 Inches, being divided by 12, gives 8 Parts: Multiply this new Multiplicand by 4 Inches, thus, 4 Inches by 8 Parts is 32 Parts, or 2 Inches and 8 Parts; fet down the 8 Parts under the Place of Parts in the upper Multiplicand, and carry the 2 Inches; then, 4 Inches by 5 Inches is 20 Inches, and 2 Inches that was carried is 22 Inches, or I Foot and 10 Inches; fet down the 10 Inches in the Place of Inches, and carry the 1 Foot; then, 4 Feet by 1 Foot is 4 Feet, and I that was carried is 5, which fet down as in the Example: Now multiply the old Multiplicand, 17 Feet and 8 Inches, by the 10 Feet, thus, 10 Feet by 8 Inches is 80 Inches, or 6 Feet and 8 Inches; fet down the 8 Inches under the 10 Inches, and carry the 6 Feet; then multiply the 17 Feet by 10, to which carry the 6 Feet, in the ufual Manner, and it makes 176 Feet, which fet down as in the Example; draw

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