PROB. II.

First, multiply all the whole feet, then all the feet and inches a∣cross, and right on, then the parts by the feet, and also the inches and parts across and right on; then add them together, and you shall have the answer in feet, long inches (that is in pieces of a foot long, and inch broad) square inches, and 8 parts of a square-inch: as for Ex∣ample.

Let a piece of board be given to be measured that is 3. 3. 5. i. e. 3 foot, 3 inches, and 5 eights one way, and 2. 3. 4. the other way, I set the numbers down in* this manner, and then right on, first as the line in the Scheme from 2 to 3 leads.

I say thus, 3 times 2 is 6, set 6 Page  [unnumbered] right under 2 and 3, as in the ex∣ample in the left page: for 6 foot, as is clear, if you consider the Scheme over the example, viz. the squares noted with f. then for the next, I say cross-wise, 2 times 3 is 6, viz. long inches, as you may percieve by the 2 long squares marked with 9 L. and 6 L. which 6 I put in the next place to the right hand, as in the example; then for the next, viz. 3 times 3 is 9 (cross wise, as the stroke from 3 to •• shews) which 9 is also 9 long inches, as the Scheme shew∣eth, and must be put under 6 in the second place towards the right hand, in the Scheme it is expres∣sed by the 3 long squares marked with L9. Then lastly for the in∣ches, 3 times 3 is 9, going right up, as the stroke from 2 threes lead you: but note, this 9 must be set in the next place to the right hand, because they are but 9 square inches, but had the product been Page  [unnumbered] above 12, you must have substra∣cted the 12 out, and set them in the long inches place, and the re∣mainder, where this 9 now stand∣eth, and this 9 is expressed in the Scheme, by the little square in the corner mark't with (□ 9.)

Then now for the Fractions, or 8 parts of an inch, first say, cross∣wise, as the longest prick line doth lead you to, 3 times 4 is 12, for which 12 you must set down 1.6, that is 1 long inch, and 6 square inches; the reason is, a piece 8 half quarters of an inch broad, and 12 inches long, is a long inch, or the twelfth part of a foot superficial; and if be 12 square inches, then 4 must needs be 6 square inches: therefore instead of 12 I set down 1.6, as you may see in the exam∣ple, and in the least long square of the Diagram or Scheme. Then do likewise for the other long square, which is also multiplied a∣cross, as 2 times 5▪ is 10, that is, as Page  [unnumbered] I said before, 1.3 as the example and Scheme make manifest, consi∣dering what I last said, and it is marked by the 2.00. But if this or the other had come to a greater number, you must have substra∣cted 8 as oft as you could, and set down the remainder in the place of square inches, and the number of 8 in the place of long inches, as here you see.

Then for the two shorter long squares next the corner, say cross∣wise again, 3 times 5 is 15, that is 1.7, because 8 half quarters an inch long, do make 1 square-inch as well as eight half-quarters a foot long made 1 long inch: therefore I set 1 in the place of square inches and 7 in the next place to the right hand, and it is expressed in the di∣agram by the small long square, and marked with * 1.7.

Then again for the other little long square, say cross-wise, as the shorter prick line leads you, three Page  [unnumbered] times 4 is 12, that is 1.4; and do by this as the last: it is noted in the Scheme by 1.4.

Then lastly, for 5 times 4, as the short prick line sheweth you, is 20: out of which 20 take the 8 s. and set them down in the last place, and the 4 remaining you may either neglect (or set it down a place further) for you cannot see it on the Rule; therefore I thus advise, if it be under 4, neglect it quite, but if above, increase the next a figure more, if 4, then it is a half, and so may be added; for note, 64 of these parts make but 1 square inch; of which parts the little square in the right hand low∣er corner of the Scheme is 20, for which I fet down 2.4, that is two half quarters and 4 of 64, which is the last work, as you may see by the Scheme and example.

Now to add them together, say thus, 4 is 4, which I put furthest to the right hand, as it were use∣less, Page  [unnumbered] because not to be expres••; then 472 are 13, from which take 8, and for it carry 1 on to the next place, or as many times 1 as you find 8, and set down the remainder, which here is 5, then 1 I carried, and 13619 is 21, from which I take 12 and set down 9, because 12 square inches is 1 long inch; then 1 I car∣ried (or more, had there been more 12 s.) and 1169 is▪ 18, from which take 12, as before, there remains 6, that is 6 long inches▪ and so had there been more 12. so many you must carry to the next place, be∣cause 12 long inches is one foot, lastly, 1 I carried and 6 is 7 foot, so that the work stands thus,

And so for any other measure Su∣perficial or Solid.

To multiply feet, inches, and 12 parts of an inch, by feet, inches, and 12 parts of an inch.

Although the usual way of divi∣ding Page  [unnumbered] the line of inches, on ordina∣ry Two-foot Rules, is into 8 parts, according to which counting the former R••• is worded, into feet, inches, and 8 parts of an inch.

Yet if every inch be divided in∣to 12 parts, o• conceived so to be, which you may easily count by cal∣ling every quarter of an inch 3, e∣very half inch 6, every three quar∣ters 9.

Then the parts between 3, 6, and 91 may be easily estimated by help of the halfquarter cut.

Then I say, the Arithmetical multiplication will be much easier, being brought to one denominati∣on after the way by decimals, and somewhat more exact, as will ap∣pear by this following example.

Suppose a Cedar▪board be 2 foot 3 inches, and 7 twelve parts of an inch broad, and 9 foot 5 inches and 8 twelve parts of an inch long, how ma∣ny superficial feet is it?

First, set down the numbers as Page  [unnumbered] is usually done in their three deno∣minations of feet, inches, and 12 parts, as in the example is made ap∣parent by the black lines and the pricked lines down right and slop∣ing from figure to figure, done so only for ease and plainness, in wording of it to a Learner.

Then proceed in the multipli∣cation thus;

1. First, 2 times 9 is 18, as the down right black line from 2 to 9 sheweth, for which set down 18 foot under the 2 and 9, as in the Example.

2. Secondly, say 9 times 3 is 27 long inches, as the sloping black line from 9 to 3 sheweth, for which 27 you must set down 2 foot in the place of feet, and 3 in the place of long inches, because 27 long in∣ches is 2 foot 3 inches.

3. Thirdly, Say 2 times 5 is 10 long inches, as the sloping black line from 2 to 5 sheweth, for which you set down only 10 long inches Page  [unnumbered] under 5 and 3 as in the the exam∣ple, because it is under 12 long in∣ches, which make 1 foot, as in the second just before.

4. Fourthly, Say 3 times 5 is 15 square inches, as the short black line from 3 to 5 sheweth; every 12 whereof makes 1 long inch. Therefore I set down 1 long inch in the place of long inches, and the 3 square inches over in the place of square inches right under 7 and 8.

5. Fifthly, Say 9 times 7 is 63 square inches, as the long pricked line from 9 to 7 sheweth, every 12 whereof makes 1 long inch, and every 144 one foot; therefore I set down 5 long inches in that place and 3 in the place of square inches.

6. Sixthly, Say 2 times 8 is 16 square inches, as the other long sloping pricked line from 2 to 8 sheweth, for which set down one long inch and 4 square inches, each in their proper places, as in the fifth last mentioned.

Page  [unnumbered]7. Seventhly, Say 5 times 7 is 35, 144 or 12 parts of 1 square inch, as the shorter sloping prick∣ed line from 5 to 7 sheweth, every 12 whereof is 1 square inch; there∣fore set down 2 in the place of square inches under 7 and 8 and 11, the parts over in a space beyond, under 144.

8. Eighthly, Say 3 times 8 is 24, 144 or 12 parts of one square inch, as before, as the other short sloping pricked line sheweth, every 12 whereof is one square inch; therefore I set down 2 in the place of square inches under 8 and 7, and no more, because nothing is over 2 halves.

Lastly, Say 7 times 8 is 56, 1728s. as the short down right pricked line from 7 to 8 sheweth, every 12 whereof makes one 144, or the 12 part of 1 square inch; therefore I set down 4 in the place of 144 & the 8 parts over in the place of 1728 s. being a place further, as in the example you see done.

Page  [unnumbered]Feet Lon. inc. Sq. inc. 1•4. 1728.

1. Then add them together, say∣ing, 8 under 1728 is 8 and no more.

2. Then 11 and 4 is 15, set down • under 144, and carry 1 to the next place.

3. One I carried and 2. 2. 4. 3, 3, under the square inches, or 12 is 15, for which I set down 3 and car∣ry 1 to the next place.

4. One I carried and 1. 5. 1. 10. 3 is 21, for which I set down 9 un∣der the long inches, and carry 1 to the next place.

Page  [unnumbered]5. One I carried and 2 and 8 is 11, for which set down 1 and carry 1 to the next place.

6. One I carried and 1 is 2, in all 21 foot 9 long inches, or 12 parts of a foot; 3 square inches, or 12 parts of a long inch; 3 144 or 12 parts of a square inch, and 8. 1728 parts of 1 long inch.

If it be a piece of Timber o• Stone; then having thus gotten the Area of the Base, then multiply that Area by the length in feet, in∣ches, and 12 parts, and the product shall be the solid content required.

As in this Example, 1. 2. 4 thick▪ 2. 1. 6 broad, and 9 foot 7 inches 6 twelves long.