The description and use of the carpenters-rule: together with the use of the line of numbers (inscribed thereon) in arithmetick and geometry. And the application thereof to the measuring of superficies and solids, gaging of vessels, military orders, interest and annuities: with tables of reduction, &c. : To which is added, the use of a (portable) geometrical sun-dial, with a nocturnal on the backside, for the exact and ready finding the hour of the day and night: and other mathematical conclusions. Also of a universal-dial for the use of seamen or others. With the use of a sliding or glasiers-rule and Mr. White's rule for solid measure. / Collected and fitted to the meanest capacity by J. Browne.

CHAP. XI. To measure any Superficies, or Solid by Inches only, (or by Foot-measure) without the help of the line, by Mul∣tiplication of the two sides.

PROB. 1.

Possibly that this little Book may meet with some that are well skilled in Arithmetick, and being much used to that way, are loth to be weaned from that way, being so artificial and exact, yet though they can multiply & divide very well, yet perhaps they know not this way, to save their di∣vision and yet to take in all the fra∣ctions together as if of one denomina∣tion: I shall begin first with Foot-measure being the more easie, and I suppose my Two-foot-rule to be di∣vided into 200 parts, and figured with 10. 20. 30. 40. 50.60.70. 80. 90. 100. And then so again to 200. as in the 3 Chap. and then the work is on∣ly thus: set down the measure of one side of the square, or oblong thus, as for example, 7. 25, and 9. 88, and multiply them as if they were whole numbers, and from the product cut off 4 figures, and you have the content in Feet, and 1000 parts of a Foot, or Yard, Ell, Perch, or whatsoever else it be. Note the examples following.

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For any kind of flat Superficies, this is sufficient instruction to him that hath read the first part; but if it be Timber, or Stone, you must thus find the Base, and then another work will give you the other side, as in Chapter 5 Problem 2. or, Multiply the length by the Product of the breadth and thickness, and that Pro∣duct shall be the content required.

PROB. 2. To Multiply Feet, Inches, and 8 parts of an Inch together without Reducti∣on, and so to measure Superficial (and Solid) measure,

First, Multiply all the whole Feet, then all the Feet and Inches, across, and right on, then the parts by the Feet, and also the Inches, and parts, across and right on; then add them to∣gether, and you shall have the answer in feet, long Inches, (that is, in pieces of a Foot long, and an Inch broad) square Inches, and 8 parts of a Square Inch: as for example.

Let a peice of Board be given to be measured that is 3. 3. 5. i. e. three Foot, three Inches, and 5 eights, one way, and 2. 3. 4, the other way. I set the numbers down in this Man∣ner, 〈 math 〉〈 math 〉 & 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 right under 2. and 3 as in the example, 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 perceive, 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, (croswise, as the stroke from 3 to 3 shews) which 9 is also 9 long Inches, as the Scheme sheweth, and must be put under 6, in the second place toward the right hand, in the Scheme it is express'd by the 3 long Squares, marked with L 9. Then last∣ly for the Inches, 3 times 3 is 9, go∣ing right up, as the stroke from the 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 a∣bove 12, you must have Substracted the 12 s. out, and set them in the long Inches place, and the remainder, where this 9 now standeth, and this 9 is express'd in the Scheme, by the little Square in the corner markt with (□ 9.)

Then now for the Fractions, or 8 parts of an Inch, first say, croswise 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 8 be 12 Square Inches, then 4 must needs be 6 Square Inches: therefore, in stead of 12, I set down 1. 6, as you may see in the example, and in the least long Square of the Diagram, or Scheme. Then do likewise for the other long Square, which is also multiplyed across; as, two times 5 is 10. that is, as I said before, 1. 3, as the Example and Scheme make manifest, considering 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 Substracted 8 s. as oft as you could, and set down the remainder in the place of Square Inches, and the number of 8 s. in the place of Long Inches, as here you see.

Then for the two shorter Long

Squares next the corner, say croswise again, Three times 5 is 15, that is 1, 7, because eight Half-quarters an Inch long do make one square Inch, as well as eight Half-quarters a Foot long made one 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 Dia∣gram by the small long square, and marked with * 1. 7.

Then again for the other little long square, say croswise, as the shorter prick line leads you, Three 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; there∣fore, I thus advise, if it be under 4. neg∣lect 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 one square Inch; of which parts, the little square in the right hand lower corner of the Scheme is 20, for which I set 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 useless, because not to be exprest; 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 re∣mainder, which here is 5, then 1 I carried, and 13619, is 21, from which I take 12, and set down 9, be∣cause 12 square Inches, is one long Inch: then 1 I carried, (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 s. so many you must carry to the next

place, because 12 long Inches is one Foot, lastly 1 I carried, and 6 is 7 Foot, so that the work stands thus, 〈 math 〉〈 math 〉 and so for any other measure Superfi∣cial or Solid.