The faithfull surveyour discovering divers errours in land measuring, and showing how to measure all manner of ground, and to plot it, and to prove the shutting by the chain onely ... / by George Atwell.

CHAP. VII. Shewing the manner of placing these upon the Rule.

FIrst, To set out the Diameter, you may take the nether part of the third column of the besil'd side, to set it on from the middle square stroke of the Rule. Then Gunther (in his Ʋse of the line of numbers in broad-measure, Prop. 11.) hath this proportion. Having the Circumference of a circle, to finde the Diameter: As 3143 to 1000, so is the Circumference, suppose it 47⌊13 to the Diameter 15: so that if you take 47⌊13 in your circle-scale, and set in that column from the middle square downward, so shall you set out 15 in that di∣stance, run that distance as oft as you can to the bottom of the Rule, which will be 4 times more, divide each of them into 3 equal parts, and the uppermost third into 5 equal, and num∣ber all the other great parts, save that with 5, 10, 15, &c. or if you will you may double 47⌊13, that is 94, 26, and take it from the circle-scale, set it there they will be 30; then half it, and they will be 15, then third it into fives.

2. To finde how to proportion the square within the circle by the Diameter. Let the Diameter be the Radius 1000, then will the chord of 90 degrees, which is the side of the square included, be the natural sine of half 90: viz. 45 degrees, the sine whereof is 707, therefore then because I would divide my scale into even sines, if therefore I take 7 times 5, that is 35, the proportion will be 707 . 1000 ∷ 35 . 49⌊50. or 49½: therefore if you take 49½ on the Diameter, and set it on the scale of chords, and divide it into 7 equal parts, and that part next the end into 5 small parts, numbring all but that with 5, 10, 15, &c. you have your scale of chords or square within the circle. Or (if you think it troublesome to divide it into 7 e∣qual parts) you may take 6 times 5, that is 30. and say 707 . 1000 ∷ 38 . 42⌊43, so then you may take 42⌊43 of the Diameter, and set on your scale of chords, and then divide each of them into halves, and each half in to 3 parts.

Otherwise thus, The content of this circle according to Archimedes is just ½ the content of the square of the Diameter. Suppose the Diameter 24, the square thereof is 576, the half whereof is 208, the root whereof is 17 ferè, then say; If 17 in chords require 24 Diameter, what shall 40 in chords, or any other even number of fives? Answer, 56½: therefore take 56½ of the Diameter, and set it in the scale of chords, which because it gives 8 times 5, first divide it into halves, then into quarters, then into eight.

3. It may also be made by this Rule of his, The area of the square within the circle is to the content of the circle as 11 to 7, so that the circle begin known, the content is thus found: ½ the Diameter multiplied in ½ the Circumference gives the content of the circle, which if you multiply by 7, and divide the product by 33, it gives the content of the square within: whereof take the square-root, and you have the side desired; therefore 19⌊8 . 88 ∷ 20 . 88⌊9, or as M^r. Wingate hath it (in Problem 33. of his Appendix to his Rule of Proportion) 225 . 1000 ∷ 20 . 88⌊9. So that take 88⌊9 from the Cir∣cumference and set it on this scale, and divide it into four fives, and this scale may be set on the lower half of the besil'd edge.

4. Having the content of the Circumference, to find the side of the square equal. Take the square-root thereof: so we found before that the Circumference being 88, the content is 616; whose square root is 24⌊40/49, that is more then 24¼. or more easily, because, as Gunther hath it, the Circumference is to the side of a square equal as 1000 the Radius to 282, therefore say, 282 . 1000 ∷ 20 . 70⌊9. Therefore take 70⌊9 of the Circumference, and set it in the scale of the square equal, it gives 20 of that scale; with which distance set out all the twenties the side will bear, dividing each 20 into four fives, and the last into five little ones, and numbring them by five as afore: and this scale may be set in the over part of the third column nexthe square edge.

Errour in round timber to take a quarter of the circumference for the square.

5. And here I must acquaint you with that monstrous errour in measuring round timber which I spake of before, which is this, to gird the piece about, and to take the fourth part for the square thereof: as suppose the piece be 80 inches about, then by this account the square should be but 22 inches: whereas in the last section we found it to be above 24¾, whereby the full fifth part of the timber is lost to the seller; which notwithstanding the most of them know to be ex∣tream false, by reason that when they have hewed it, they make a great deal more of it, then they did before it was hew∣ed. But what is their excuse? Even this they say, That will scarce pay for the hewing, and it is but sap and bark. I answer, The goodness or badness of any thing is considered in the price; but neither in the measure nor the manner of measuring. I have seen a sack of fine seed, white wheat, sold for ten shillings a bushel, another of grey wheat at seven, sold the same day all to one man: yet he had no more measure of the course grey, then of the fine wheat. Secondly, In that they say, They had need have that for hewing: I say, They never hew what they rend to laths, pales, rails, plow-timber, cart-timber, wheel-timber, boles, trenchers, dishes, spoons, and infinite other, which they rend, and sell sap and all. Thirdly, When they do hew any timber, they leave it so wany, that (in Cambridge-shire especially) they leave it nearer round then square; and yet allow nothing for the wanes: so that in all other things, whether sold by weight or measure, the buyer is to have the draught, though it be but in an ounce of pepper, in this he must want of his measure, and that no small matter; for they seldome hew nigher to square in this Countrey, then that the four wanes are as broad as the four flats, all which are equal to a square piece of the breadth of one of those wanes; & although those wanes be less in some places then in other, yet will they be of no service so deep as the deepest wane goes. And what sense or equity is there, that in buying they

should desire so much over-measure, and yet in selling it hew∣ed sell so much short, as in buying? Hath not he that buyeth wane-timber, that the wanes run not streight, as much need, and as much reason, to have allowance for the wanes, and to have the knots and bark left on them for hewing, as you to have the fifth part and more, and yet never hew a great deal of it at all? Besides, you have a trick, when you buy round-timber with the bark on it, be it thick or thin, you will cut a notch round about the piece in the middle of the block, sometimes deeper then the bark, saying, That is but a boin: now you buying by measure, what right have you to the bark, which you measure not? yet when it is hewed, they that buy it must be content with air instead of timber. And yet fur∣ther, I have known a Wheel-wright, that used to buy all his timber by the foot of fourteen inches every way to the foot, and to girdle it, and to take the fourth part for the square; thus did he over-reach the sellers, who thought it to be but a seventh part more then ordinary, and that he gave a penny or two pence more in a foot then others gave, they thought them∣selves well enough; whereas (poor simple fools!) they sold above two foot for one.

6. If you buy round timber that is ordinarily taper, little or much, then you will be sure to gird it in the middle, or nearer the little end, whereby you gain no small matter.

Lastly, How common a thing is it with Wood-mongers, to have one Rule to buy by, & another to sell by: one a quarter of an inch too long▪ another as much too short? And great pity it is, that considering there are so many abuses in measuring land and timber, it is not a whit looked into, whereas in all other things sold by weight or measure the abuses are punished by the Clerk of the market.

Now for correction of this false measure in round timber; committed by this way of taking the fourth part for the square, if it be a perfect Cilinder, and not taper, you may help your self by this Table, taken out of M^r. Stirrup's Plain-scale, or Carpenters new Rule, page 60, which you may draw into a

scale, as you do for square timber or board-measure; all but the first seven inches, which are under-measure, and set those 7 in four columns, between the two Tables of board and timber under-measure.

Squar. Inch.	Feet.	Inch.	Cent.	Squa. Inch.	Inc.	Cen.	Squa. Inch.	Inc.	Cen.
1	113	1	71	11	11	22	21	3	11
2	28	3	42	12	9	42	22	2	80
3	12	6	85	13	8	3	23	2	56
4	7	0	85	14	6	92	24	2	35
5	4	6	30	15	6	3	25	2	17
6	3	1	71	16	5	30	26	2	0
7	2	3	70	17	4	69	27	1	86
8	1	9	23	18	4	19	28	1	75
9	1	4	76	19	3	76	29	1	61
10	1	1	57	20	3	39	30	1	51

The use of this Table is thus.

Girt the piece about, and take the fourth part for the square, as if it were the true square, and therewith enter this Table; and it gives the feet, inches, and Centesmes required forward to make a foot forward at that false square. So 44 inches circle gives 11 inches for the fourth part, which in the Table gives 11 inches, 22 Centesmes, forward to a foot-square of timber. Or else having taken the Circumference with a nea∣led wier, and there made a twist, and measured the number of inches about, take off so many with your compasses, and ap∣ply that wideness to the scale of the square-equal, and you have the square you must measure it at. And because as I said before, that to hew a log for boards, the best way is to hew it eight-square, both for saving timber, and to have all the boards streight-edged; so neither shall the sawyers be paid for more then they saw, nor he that buieth the boards or the block it self, want, or have too much: we will now therefore give

you one rule whereby to measure all equal-sided timber, so that it be not taper, how many sides soever it hath. First, finde the centre of your piece, and measure the semi-diameter thereof to the middle of one of the equal sides; then add all the sides together, multiply half thereof by the semi-diameter: so have you the content of the base, and that multiplied in the length gives the content of the piece. So in the figure the 8 sides are ten

[illustration]

a piece, that is, 80; the half whereof is 40; the semi-diame∣ter or perpendi∣cular AB is 1••, that multiplyed by 12 makes 480, which is the content of the base, that is, one inch sawed off of the end of the piece. Then if either you multiply 480 by the inches of the length of the piece, and divide the product by 1728, you have the content of the piece. Or else you may extract the square-root of 480, which is 22 ferè, and then measure it, as if it were 22 inches square. And thus may you measure all manner of timber, not taper, by measuring one inch at the end, as if it were land: then extract the root, and measure is as if it were so much square.