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.

¶ An Appendix to my Faithfull Surveyour.

WE have, in the book it self spoken of measuring such things, as are measured by observing Instru∣ments, as the Pandoron, plain-Table, Quadrant, Quadrat, Theodelete, Circumferento••, &c; viz. of measuring of land, taking of Altitudes and Di∣stances, taken by the chain: here we will speak of such super∣ficies as are done by a two-foot-rule, as board, glass, pave∣ment, wainscot; and of solid, as stone and timber: forbear∣ing those things, that seldome, or never, come in question; as globes, regular bodies, and the like. First, Because land-measure and those seldome meet together in one man; Se∣condly, Neither would I have the book to be of two big a price; and Thirdly, Because my little time I have, hath need to be spent to the best advantage for the common good.

CHAP. I. Of making the Rule.

FIrst, I would have the Rule, (whether it be of box, or of brass; whether joynted in the middle, or streight out) to be just two-foot-long by some standard of brass, kept by the Clerk of the Market and not, as I have seen some; that have been half an inch too long. Let it be an inch and an half broad at the least, and a third part of an inch thick with a square stroke struck round about it just in the middle of the length thereof. Let one edge be besild off: which serves that if you have occasion to draw lines with a pen, if you turn that side downward, you need not fear blotting: if your rule chance to be blackt with inke, if you rubb it well with sorrel, that will fetch it out. Through the midst of this besill strike a Gage-stroke: an another along the midst of the other edge:

divide the rest of this side, beside the besill, into eight equall parts with seaven Gage-strokes. In the 4 next co••umnes save one to the besill, you may place all the under-measure of this Table of board-measure following, which will not fall in a scale upon the rule, viz. all inches, halves, and quarters from one inch to six, or if you will to ten inches, in small spaces the inches of the breadth of the hoard, in the column next save one to the besill: the feet required to a foot foreward at the breadth in the next▪ the odd inches in the third and the Gen∣tesmes in the fourth. And adjoyning to this Table toward the middle of the Rule, in the first of those four columnes se•• one inch divided into ten equall parts, and each of those into halves, and each of those halves into five; or suppose them so divided: so is it divided into 100 parts or Centesmes: from which inch you shall take off all your Centesmes with your compasses, that are to be set in any of your scales.

For making the scale of board-measure.

Before you can make this scale, you must have one column, on the otherside the Rule, next the besill, parted into three small parts with Gage-strokes, and divided in the middle of the length of the rule into two equall parts or feet: whereof divide one of them into ten equall parts, and each of them into ten more, and each of them suppose at least to be divided into ten other; so shall that foo•• be dvided into 1000. and this Gunther calleth foot-measure: which must be reckoned both wayes, first from the beginning of the rule to the middle, thus, 1, 2, 3, &c. and backward again, and thus, 11, 12, 13, &c. and because the other foot makes ten of these inches, and these ten make twelve of them, therefore divide the other foot into twelve equall parts or inches, and each inch into eight parts, and number it from the end toward the middle with 1, 2, 3, 4, &c. but from the middle to the end with 13, 14, 15, &c. and this he calleth inch-measure. By help of this inch-line and the inch aforesaid, and by help of your Tables for board and timber-measure, are made your scales for board and timber-measure. And this Table of board-measure

is thus made: First, for all whole inches divide 144 by the inches of the breadth, and you have the inches forward to a foot. If any thing remain after division, it is the Numerator of a common Fraction, whose Denominator is the Divisor; to which remain annex two ciphers on the right hand, and di∣vide again by the same Divisor, and you have the Centesme de∣sired. Example.

Let a board be seven inches broad, I desire to know how many inches forward makes a foot. Divide 144. by seven, it gives twenty inches; or one foot eight inches ••/7. Now to bring ••/7 into centesmes, annex two ciphers to the remain four, it makes 400▪ which divide again by seven, it gives ••••/100. But for half-inches reduce the breadth into an improper Fraction, as 6½ is 1••/2; then multiply 144 by the Denominator 2, it gives 288: so that you must always divide 288 by the Numerator, or number of half-inches of the breadth of the board, which is 13; so have you 22, or one foot, ten inches, 15 centesmes. But if your breadth be an odd quarter, or three quarters: First, reduce it into quarters, and divide 576 by it: so •• ¼ is 27 quarters, therefore divide 576 by 27, it gives 21 inches; or one foot, nine inches, 9/27, or 33 centesmes. The Table fol∣loweth.

A Table shewing how many feet, inches, and centesmes of inches forward are required to make a foot of board mea∣sure at all breadths, both whole inches, half-inches, quar∣ters, and three-quarters, from one inch in breadth to 36 inches.

Quar. Board.	feet.	inch.	cent.	Quart.	feet.	inch.	cent.	Qu.	inch.	cent.	quar.	inch.	cent.
1 0	12	0	0	8 0	1	6	0	15	9	60	22	6	55
1	9	7	20	1	1	5	46	1	9	44	1	6	47
2	8	0	0	2	1	4	94	2	9	29	2	6	40
3	6	10	29	3	1	4	46	3	9	14	3	6	33
2 0	6	0	0	9 0	1	4	0	16	9	0	23	6	26
1	5	4	0	1	1	3	56	1	8	87	1	6	19
2	4	9	60	2	1	3	16	2	8	73	2	6	13
3	4	4	36	3	1	2	77	3	8	57	3	6	6
3 0	4	0	0	10 0	1	2	40	17	8	41	24	6	0
1	3	8	31	1	1	2	5	1	8	32	1	5	94
2	3	5	15	2	1	1	76	2	8	22	2	5	88
3	3	2	40	3	1	1	35	3	8	12	3	5	82
4 0	3	0	0	11 0	1	1	9	18	8	0	25	5	76
1	2	9	88	1	1	0	80	1	7	81	1	5	70
2	2	8	0	2	1	0	51	2	7	78	2	5	65
3	2	6	31	3	1	0	25	3	7	68	3	5	59
5 0	2	4	80	12 0	1	0	0	19	7	58	26	5	54
1	2	3	41	1	0	11	76	1	7	48	1	5	48
2	2	2	18	2	0	11	52	2	7	39	2	5	43
3	2	1	4	3	0	11	29	3	7	29	3	5	38
				Qu.	Inch.	Cent.
6 0	2	0	0	13	0	11	8	20	7	20	27	5	33
1	1	11	4		1	10	87	1	7	11	1	5	28
2	1	10	15		2	10	67	2	7	2	2	5	24
3	1	9	33		3	10	46	3	6	94	3	5	19
7 0	1	8	57	14		10	29	21	6	86	28	5	14
1	1	7	86		1	10	11	1	6	78	1	5	11
2	1	7	2••		2	9	93	2	6	69	2	5	5
3	1	6	58		3	9	76	3	6	62	3	5	1

Q.	I.	C.	Q.	I.	C.	Q.	I.	C.	Q.	I.	C.
29	4	97	31	4	65	33	4	36	35	4	12
1	4	93	1	4	61	1	4	33	1	4	9
2	4	89	2	4	58	2	4	30	2	4	6
3	4	84	3	4	54	3	4	27	3	4	3
30	4	80	32	4	50	34	4	24	36	4	0
1	4	76	1	4	46	1	4	21
2	4	73	2	4	43	2	4	18
3	4	69	3	4	39	3	4	15

Now to place this Table upon the rule, divide the second, third, fourth, and fifth columns next to the besill, at one end into small squares that may hold two figures a piece, in which set over-most the inches of the breadth, in the second the feet required in length, at each inch, half inch, and quartern. In the next the odd inches, and in the next the odd centesmes: and this you must do to six inches, you may do it to ten inches if you will. Then at the end of ten inches, set one inch divi∣ded into ten equal parts, and each of them into halves, and suppose each half into five, so will it be supposed to be divi∣ded into an hundred parts, as before. Then from six inches to 36 you shall set all in the column next the besill, with small strokes, after this manner: First, I begin with six inches and a quarter, to which I finde in the Table there belongeth one foot, eleven inches, four centesmes, that is eleven inches, four centesmes from the middle cross stroke of the rule. But be∣cause my compasses will not reach so far, I onely take 56 cen∣tesmes from the former inch, which makes it just two foot from the same end, which I set the under measure at.

Another example let be 9¼, for which I finde in the Table one foot, three inches, 56 centesmes. First, I take with my compasses 56 centesmes from my inch of centesmes, and prick it down upon a line upon a paper. Also with my compasses I

take three inches in the foot-line of inch-measure on the other side of the Rule: set that distance also on the paper at the end of the 56 Centesme in the same line; then take with your com∣passes the whole length of both, set one foot in the middle-cross-line of the Rule, and in the said scale, and the other to∣ward the beginning of the Rule, and it gives the length corre∣spondent to nine inches and ••/4, from the stroke to the end of the Rule. Thus do with all the rest; marking each whole inch with its proper number to 24, also 30, and 36.

And now, before we proceed to shew you the making of the Table of timber-measure, we will first shew the measure of boards.

CHAP. II. Of measuring of boards with the Rule.

THere are divers ways of measuring of boards: of which the fundamental way is this; 12 inches in length, and 12 in breadth, that is twelve times twelve, or twelve inches square, which is 144 inches, make a foot of board: therefore multiply the inches of the length of the board by the inches of the breadth, and divide the product by 144, you have the content in feet. If any thing remain, divide it by twelve, it gives the odd inches, or twelve parts of a foot: for an inch is the twelfth part of a foot, let the foot be what it will. Example.

Let a board be 13 foot five inches long, that is 162 inches long, and nine and an half broad, these multiplied give 1529 and an half, which divided by 144, give ten foot, & 89 square inches and ½ remains, which divided by 12 is 7½ ferè inches of board. Secondly, If you multiply the length in feet, 13 feet 5 inches, by the breadth in inches 9½: first, 9 inches by 13 foot, is 9 foot 9 inches; & half of 13 is 6½, and 6 square inches; and 9 times 5 inches is 45 square inches; and half five inches is two and an half square inches. First then, add all your inches together, 45, 6 and 2½ make 53 and ½, which divided by 12, gives 4 board inches, and 5½ square inches, or half a board

inch feré. Now add these 4 inches to 9 and 6 inches, they make 19 inches, that is, one foot, seven inches, to which add 9 foot, it gives ten foot, seven inches ½ ferè, just as afore: and both those ways are performed by any common Rule that ••ath no board-measure on it. Hence then is discovered this errour, that if a board be nine inches broad, to take 15 inches forward to make a foot, that is so much more then twelve, as nine is less, where∣as our Table saith you must take 16, is a false way: for nine times 15 is but 135, which wants nine square inches of 144, and is always the square number of half the difference of nine and 15 equally distant from 12, whose square is 9. So like∣wise 8 and 16 being multiplied make 124, which wants 16 of 144: and because they are equidistant from 12, and their half difference is 4, therefore their product is less by sixteen, the square number of four, then the square of twelve.

3. A third way of measuring board is by this rule, Mea∣sure the breadth of the board; if it be less then six inches, your Table of under-measure will shew you how much forward you must take to a foot forward. If it be broader, and under 36 inches, then the strokes on your scale give it.

4. Some measure all the breadths of the boards with a line, then stretch the length on a block, and so measure the breadths of all the stock at once, and then measure the length of a board, then multiply the length in feet and parts, by the breadth in feet and parts: So suppose the breadth of all the boards is ten foot, nine inches, and the length 154 inches, instead of nine inches, I take ½ ¼ of a foot, and instead of four inches I take ⅓ or ¼ one inch, and the work will be thus, and it makes 164 feet ¾, 1 inch and an half. 〈 math 〉〈 math 〉

And this is a very good way in case a block be hewn eight-square, before it be sawn: which if it be fit for boards, it is pitty it should be hewn any other way; so will it be no loss of timber, the boards will be all streight-edged. If it be sold in timber, and measured as eight square,

(as shall be shewn) there will be no loss either to buyer or seller.

CHAP. III. Of making of a Table of timber-measure for square timber, to make the scale of square timber-measure by: as also the under-measure.

FIrst know that a foot of timber is twelve inches every way breadth, length and thickness, and therefore conteineth 1728 square inches, for 12 times 12 is 144, that is, a foot of board or a superficies, and twelve foot of board make 1728 inches; therefore to proceed to the Table. First, For whole inches: square the square of the piece, that is, multiply the square by it self, and by that product divide 1728. Example. Suppose the piece be 8 inches square, the square of 8 is 64, by which divide 1728, it gives 27 inches, or two foot, three in∣ches But if you have odd half-inches, then you must reduce as before all your inches into half-inches, or an improper Fraction, by whose Denominator (which will always be 4) multiply 1728, it gives 6912, which must always be divided by the Numerator of the Fraction. Suppose the square given be 6½, that squared is 42¼ which reduced is 169 quarters; by which 169 divide 6912, it gives 46 inches, or 3 foot 4 inches ninety Centesmes. Again if the square be of odd quarterns or ¾ you must work as before, and then your divide••t will be 16 times 1728, that is, 27648. Example. Let your square be 6¾, that squared is 45 & 9 sixteenths: which reduced into 16 parts by multiplying 45 by 16 and adding 9, it gives 7 19 sixteenths. Therefore divide 27648 by 729 it gives, 7 inches, or 3 foot, 1 inch, 92 Centesmes.

Here followeth the Table of timber-measure.

Inch squar.	feet.	inch.	cen.	inch squar.	feet.	inc.	cent.	Inc.	Inc.	C.	Inc.	Inc.	C
1 0	144	0	0	8 0	2	3	0	15	7	68	22	3	57
1	92	1	92	1	2	1	39	1	7	43	1	3	49
2	64	0	0	2	1	11	91	2	7	19	2	3	41
3	47	0	24	3	1	10	57	3	6	97	3	3	34
2 0	36	0	0	9 0	1	9	33	16	6	76	23	3	27
1	8	5	33	1	1	8	19	1	6	54	1	3	20
2	23	0	48	2	1	7	14	2	6	35	2	3	13
3	19	0	60	3	1	6	25	3	6	16	3	3	6
3 0	16	0	0	10	1	5	28	17	5	98	24	3	0
1	13	7	55	1	1	4	44	1	5	81	1	2	94
2	11	9	6	2	1	3	67	2	5	64	2	2	88
3	10	2	88	3	1	2	95	3	5	48	3	2	82
4 0	9	0	0	11	1	2	28	18	5	33	25	2	76
1	7	11	67	1	1	1	65	1	5	19	1	2	71
2	7	1	33	2	1	1	6	2	5	5	2	2	66
3	6	4	75	3	1	0	51	3	4	91	3	2	61
				Inch.	Inc.	C.		In.	In.	C.	In.	In.	C.
5 0	5	9	12	12	12	0		19	4	78	26	2	56
1	5	2	69	1	11	51		1	4	66	1	2	51
2	4	9	12	2	11	6		2	4	55	2	2	46
3	4	4	26	3	10	63		3	4	43	3	2	41
6 0	4	0	0	13	10	29		20	4	32	27	2	37
1	3	8	23	1	9	82		1	4	21	1	2	33
2	3	4	89	2	9	48		2	4	11	2	2	29
3	3	1	92	3	9	14		3	4	1	3	2	25
7 0	2	11	27	14	8	82		21	3	92	28	2	21
1	2	8	88	1	8	52		1	3	83	1	2	17
2	2	6	72	2	8	22		2	3	74	2	2	13
3	2	4	77	3	7	90		3	3	66	3	2	9

In.	In.	C.	In.	In.	C.	In.	In.	C.	In.	In.	C.
29	2	6	31	1	80	33	1	59	35	1	41
1	2	2	1	1	77	1	1	56	1	1	39
2	1	99	2	1	75	2	1	54	2	1	37
3	1	95	3	1	72	3	1	52	3	1	35
30	1	92	32	1	69	34	1	49	36	1	33
1	1	89	1	1	66	1	1	47
2	1	86	2	1	64	2	1	45
3	1	83	3	1	61	3	1	43

To place this Table on the Rule.

Begin at the other end of the Rule taking those 4 columns next the thick edge save one, and divide them into little spaces, as you did for board-measure, setting on them all the under measure to 8 inches and an half square, yet you may do it to 12 inches, if you will; setting the square inches of the block in that column next save one to the edge: then the feet requi∣red to make a foot forward in the next: then the odd inches in the next to that, and the Centesmes in the last of the 4. Then from 8 and ½ to 36 you may take off your inches from your line of inch-measure, and your Centesmes from your inch of Centesmes, as you did in board-measure, and place it back∣ward or forward, according as it shall be more or less then a foot.

CHAP. IIII. Of measuring solids, as stone, timber, &c. and first of square timber.

FOr measuring all kind of solids the fundamental or gene∣ral way is to multiply the inches of the breadth by the inches of the depth, and that product by the inches of the length, and divide the last product by 1728. This is so plain, it needs no example: and this is the best way for stone of all other.

2. A second way of measuring square timber is by this Ruler. Having the square of the piece given look on the Rule, and see how often you finde the length required at that square between that and the end of the Rule in the length of the block, so many foot of timber is in that block.

To finde the true square of a piece broader one way then another.

But to finde the true square of the piece, multiply the breadth by the depth, and from the product extract the square-root.

As let the breadth be eight, and the depth 14, these multi∣plyed make 112, whose square root is 10 1••/21, according to which square you must measure the piece. Which disproveth a common errour; which is this, To add both sides together, and to take ½ thereof for the square: for so 8 and 14 make 22, the half thereof is 11. And although there seemes but small difference, viz. less then ½ an inch between their num∣bers or roots 10 12/21 and 11: yet between their squares there is no less then 9 inches difference, for 11 times 11 is 131, but 8 times 14 is but 112.

3. Now therefore because every Carpenter cannot extract the square-root, and to them that can do it, it is but a slow way: and thirdly we never set any scales of timber-measure upon Rules, but for inches, halves and quarters: take this for the best way of all other, where there is such difference of the sides measure it first that false way, then take out of it always a square piece of ½ the difference of the sides, quite through the block; so in our example 8 and 14, their difference is 6, the ½ thereof is 3: therefore take a piece of 3 inches square through the length of the block, for that 3 squared gives 9▪ which is the difference between the square of it and the rectangle of 8 times 14.

CHAP. V. Of round t••mber.

BEcause to every circle there belongeth 3 squares, first the square without the circle, or the square of the diame∣ter; secondly, the square equal to the circle, not in Peripherie, but in the area; for if the area of a circle of a mile round, and a mile about in a square be compared, we shall finde the square to contain just 40 acres, whereas the circle of the same Pe∣ripherie containeth 50 acres, 3 roods, 25 poles 5/11; and thirdly the side of the square within the circle: therefore we will first shew the manner of making these 4 scales, and then the mea∣suring of round timber: yet before we shew the making of them our best way is to take Virgil's advice, and to do as he doth with his Bees.

Principio sedes apibus statióque petenda.

So before we shew the making of them we will first finde out a seat for each of them, and then the making of them one after each other. First; in the beginning of the first cha∣pter we shewed that we would have one of the edges on one side besild off: and the rest of that side divided length wise into eight equal columns with 7 Gage-strokes upon the besill, ½ the length of the Rule, you may set a scale of 20 in the inch dividing each inch into halves and quarters. Numbring each half-inch with 10, 20, 30, &c. save that half-inch next the beginning, which must not be accounted for any of the tens: but that must be divided into ten equall parts by it self, to take the odd inches above even ones, that any round block or circle is about.

Besides this, you have three other scales that are for round measure, that shew the three squares belonging to the circle: and any of these four being known, all the rest are known onely by taking the number thereof upon its proper scale with your compasses, and apply that distance to the scale proper to the thing desired: and these three scales for these squares are

one for the Diameter, or side of a square without the circle, and that each side thereof toucheth the circle. Another is the side of a square within the circle, or of the chords of 90 degr. and the other is a side of a square, whose content is equal to the content of a circle. For Example. Let a block be girded about with a nealed wyer, and then that wyer laid along upon the block, being found to be 88 inches, I set one foot of the compasses in 80 of the said circle scale, and the other foot in 8 of those 10 odd parts next the beginning of the Rule, rec∣koned from ten upward, being the contrary way to the other 80. If then you desire to know the Diameter of the circle, or side of the square including the circle, you shall finde it just 28 inches, by setting one foot of the compasses in 25 of the Diameter scale, and the other will fall in three odd parts, which added make 28: for all these three last scales must be divided into fives, and numbred with 5, 10, 15, &c. and five odd ones above, at the beginning. Likewise if you apply the same wideness of the compasses to the scale of the square within the circle, that is, to the square, that a block being round will be, being hewed just to the four edges: then set one foot of the compasses in one of those great divisions by fives, so that the other may fall amongst the odd small divisi∣ons, and it gives you 19¾ feré.

And lastly, if you apply the same wideness of the compasses to the scale for the square equal, setting one foot in the great divisions, so that the other may fall in the five odd small ones, it gives 24 and about ⅔.

And in like manner if any of the other three scales be given, as if the Diameter 14 be given; if you take 14 upon the Diameter, and carry that to the circle; it gives 44; if to the square equal, it gives about 12⅓, and so of the rest.

CHAP. VI. Of the proof of these scales by Arithmetical calculation.

FIrst, for the circle-scale, that needs no proof, so that it be truly divided: for that is the basis, on which the other are built; or scale, by which they are made.

Secondly, For the Diameter Archimedes gives this rule, Mul∣tiply the Circumference by seven, and the product divide by 22, so have you the Diameter: so on the contrary. Thus our circle 88, multiplied by seven, gives 616, which divide by 22, quoteth just 28, as afore.

Thirdly, For the square within the circle this is the rule. The square without the circle is double in content to the square within. Or thus, The content of the square within the circle is to the content of the circle as 7 to 11: First, therefore by the content of the square without, we found the Diameter, or side of the square to be 28, that squared or multiplied by it self is 784, the content thereof. Therefore the content of the square within is but ½ 784, that is, 392. whose square-root is 19 31/39, as afore. Secondly, by the content of the circle: for which Archimedes saith, half the Diameter multiplied by half the Circumference gives the content, so 44, the half of the Circumference, multiplied by half the Diameter 14, gives 616, the content of the circle. This therefore multiplied by seven, makes 4312, which divided by eleven gives 392, just as afore.

Fourthly, For the square equal to the circle, having by this last rule found the content of the circle to be 616, we need but extract the square-root thereof, which is 24 40/49, which doth discover a most monstrous, and a most gross errour in measuring round timber, of which hereafter.

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.

CHAP. VIII. Of taper-timber, whether Conical or Pyramidal.

FOr such kinde of timber of either sort, measure it as if it were a whole Cilinder or Prisme, that is, First, finde the area of the base, and multiply it by the whole length, thus; Let a Prisme be four-square, the side 12, the area of the base is 144, and suppose the length 100, these multiplied make 14400. But by the Corollary of the 7^th Prop. 12. lib. Euclid. every Pyramis is the third part of a Prisme, having the same base and altitude: therefore divide 14400 by 3, it giveth

4800 the content of the Pyramis. But suppose it be an imper∣fect Pyramis, that runs not to a point, but hath his top cut off: you shall then continue out the sides to a perfect Pyramis, by plotting it in paper, or else finde how much it wants by the Rule of three. Example.

The side of the base being twelve, the length of the piece fiftie, and the side there is six, so that there is six lost in fiftie; but the whole side of the base is but twelve, whence take six, six resteth. Then say 6 . 50 ∷ 6 . 50. and 50 and 50 make an hundred, as before. Now then for this little Pyramid, the side or Diameter of the base thereof being six, whose square is 36, the third part whereof is twelve, that multiplied by 50, gives 600, the content of the lesser Pyramid. Subtract this perfect Pyramid out of the great perfect Pyramid 4800, rests 4200, the imperfect Pyramis. And the reason, that holds be∣tween the Prisme and Pyramis, holdeth also between the Cilin∣der and Cone, Prop. 10.12. Euclid. Every Cone is the third part of a Cilinder, having the same base and altitude.

Of the Cone.

Let us now suppose a Cone also divided in length into 50 and 50, the greater Diameter at the base to be twelve, and six in the middle. First, to finde the Circumference to 12, the Di∣ameter: 12 multiplied by 22 is 264, that divided by 7 is 37 5/7, the Circumference. Then multiply half 37 5/7 (that is) 18 6/7 by half the Diameter, (that is) six, it gives 115 5/7, the greater area, which multiplied by 100 the length, it gives 11514 2/7 the Cilinder, the third part whereof is 3838 2/21 the greater Cone, Now for the lesser, the Diameter is six, multiply it by 22, it is 132, that divided by seven, is 18 6/7 the base, which multiply by the length 50 is 942, the third part thereof is 314 2/7 the lesser Cone.

Now take 314 2/7 out of 3838 2/21, resteth the imperfect Cone 3520, which is almost twelve times as big as the lesser. Or, if you rather desire 12 and 6, the bases of the Pyramis, to be the sides of the square within the circle, as there they are, and then to see their dimensions: then first, if twelve be a side

of a square within the circle, since the content, or square there∣of, is but half the content of the square of the Diameter: therefore double the square thereof, and out of the double extract the square root, and you have the Diameter: so 12 squared is 144, that doubled is 288; whose square-root is 17 ferè, the Diameter.

Now to finde the Circumference, multiply 17 the Diame∣ter by 22, facit 374. that divide by seven, it quoteth 53 ••/7 the Circumference: then multiply half the Circumference 26 5/7 by half the Diameter 8½, it gives the area of this base 227 ••/14, which multiplied by 100, the length, gives 22707 ••/7 the Ci∣linder, which divided by 3 gives the great Cone 75695½. Like∣wise for the lesser square within, which is six, the square is 36, that doubled is 72, the square-root whereof is 8½ ferè, the Diameter. Multiply 8½ by 22, it gives 187; which divided by 7 gives 26 5/7 the Circumference, then multiply half 26 5/7 (that is) 13 5/14, by half 8 & an half (that is) 4¼, and you have 56 577/879 or ⌊72 ferè, the content of that area; which multi∣ply by 50 the length gives 2835: the third part thereof is 945, the lesser Cone. Take this lesser 945 out of the greater 7569, resteth 6624, the imperfect Cone: So that the imperfect Cone is more then seven times as big as the little one.

The discovery of severall errours in measuring the Pyramid and Cone: and first of the Pyramid.

Some hold that to be true, To add the areaes at both ends together, and multiply the 1 half thereof by the length of the piece, as in our example the area of the great end is 144, and the little end nothing therefore half 144 (i. e.) 72 multiplyed by 100 is 7200, but it should be but 4800: it is too much by 2400.

A second errour is to take the area at the third part from the great end, as in this figure, at C and C, but there the square or side is 8, and the square number or area thereof is 64, which multiplied by 100 is 6400, too much by 1600.

A third errour is to take the square in the midst of the piece,

as at B and B, where the side is 6, the

[illustration]

area 36: that multiplied by 100 the length gives 36••0, which is too little▪ for take 3600 out of 4800, the diffe∣rence is 1200; a just quartern lost of the timber to the seller; so that it falleth near the middle between B and C, where it is 7 inches, for that gives 5900, yet there it is too much by an hundred.

Secondly in the Cone.

The common practise is to gird it in the middle, and to take the fourth part for the square. In measuring the ci∣linder, there was more then the fift part lost to the seller: but here that it is taper also, is a more intolerable loss. For if in the square Pyramid was lost a full quartern onely by reason of taper∣ing: what will here be lost where two such errours combine in one to wrong a man? The Circumference in the midst of the piece is 26 5/7, the fourth part thereof is 6¾, which squared is 45½ and that multiplied by 100 makes 4556 ••/4, which taken out of 75 9•• there is lost to the seller 3013, which is almost one half thereof. Yet this goeth so for currant in all places, that he that con∣tradicts it is scorned as a fool, and ac∣counted as a knave.

CHAP. IX. Of the making of four other lines on the flat-sides, whereof three are M^r. Gunthers lines, of numbers, sines, and tangents; and instead of the Meridian-line, which is onely usefull for Navi∣gation, whereof Carpenters make little or no use, we have added a sextant of chords.

ALthough M ^r. Wingate (in his book called The Rule of Proportion,) hath set down the making of them: yet for that he hath done them after another manner then here is shown, neither will an ordinary Rule bear all those lines, we will therefore content our selves with M ^r. Gunther's, & the line of chords onely. You shall divide the rest of the Rule be∣side the columns of feet & inch-measure before spoken of, into four other great columns, and divide each of them into two equal, and one of them into two also; so the great shall be for figures, the other 2 for strokes. These two of M^r. Gun∣thers you may set in the three middle columns, and the line of chords on the other outside.

First, for making the line of numbers.

I told you before that I would have you strike a stroke round about cross the Rule, I would also have another at each end of the Rule so close as possibly you can, onely to set one point of the compasses on. Then first set out your great division in each foot; viz. the thousands, if your number consist of four figures, or howsoever they are to be the left hand figures of any number, as 3 in 3 32.346.3654.37046, &c. and must be marked with the 9 digits in either foot, and the first last and middle-most with one, so that you may understand as many ciphers with it as shall be requisite, so that it may signifie 1.10.100.1000. and then if one signifie 10 the next two will naturally signifie 20, but not always. Now to take and set the number 2 in his right place, take a Table of Logarithmes of absolute num∣bers, and look either the Logarithme of 2.20. or 200. and

take the three next figures to the Characteristick, which are 301: then with your compasses take 301. viz. three inches, no tenth part of an inch, and 1/10 of a tenth part or Centesmes of an inch, and set one foot in the nether-most cross stroke, where you set the first one, and turn the other upward in the same column, and there set your 2 likewise with the same numbers, set one foot in the middle cross stroke where you set the middle one, and turn the other upward toward the up∣permost one, and there set your 2 also: likewise, do with 3 whose Logarithme is 477 (id est) 4 inches, 7 tenths, 7 Cen∣tesmes: also with 4. And these figures for the making of this line we will call hundreds, the next subdivision tens, and the least Centesmes. But now because we will suppose your com∣passes will not well reach beyond the figure 4, whose Loga∣rithme is 602, that is above 6 of those inches: therefore first, let us set on the tens so far on both feet, and then the rest of each foot afterward. Next set out each fifth tenth so far: be∣cause you must mark them with longer strokes, then each sin∣gle ten: so then you must not account the next of those fifths to 1 as 5. (for then you will account the one for nothing) but you must account it for 15. or 150. and so take the Loga∣rithme thereof, which is 176. Likewise 25, or 250, is 398, which you must take with your compasses, and set in their places in in both feet, and in like sort shall you do with all your single tens; accounting that next •• not for 1, nor 2, but for 11. Or instead of taking them off with your compasses, strike out all the first foot with a fine small striking squire of brass, lay∣ing it upon the Log. in the line of foot-measure, and then set out the other foot with your compasses by this.

Now for the rest of each foot, look out the Logar. of your numbers, and take the distance between it and the middle cross-stroke, and with that wideness set one foot in the upper 1, and where the other falls, there is the place of that number. Ex∣ample. I would set out 70, the Log. is 845; I take the distance between it and the middle-stroke of the Rule, or the Arith∣metical complement of it, 154, and set it both from the upper

stroke and middle-stroke downward, and you set out seventy. But your over-foot may bear unites to 20, and from thence to 40, divide each tenth into five, and from thence to the end into two.

To make the line of sines.

First, you must know that neither the line of sines, nor tan∣gents, enter the Rule till 35 minutes: where you see the two next figures to the characteristick 8, are both ciphers; there also the characteristick changeth from 7 to 8: for your chara∣cteristick shews what foot you are in: therefore since we rec∣kon the minutes onely by tens, our first number or division up∣on the Rule will be at 40 minutes of the first foot, shewn by the characteristick 8: for 9 is the last, and therefore belongs to the last foot; so that whereas you see that the Log. of one mi∣nute hath 6 the characteristick, & 463 the three next figures: therefore one minute would be above a foot and half before the entrance on the Rule, and likewise would the first minute of the tangents be. Now the Logar. of 40 minutes hath beside the characteristick 8 the three first figures 066 feré: therefore take off 0 inch, 6 tenths, and 6 centesmes, or 5 centesmes, and 7 millesmes, if you ca•• ghuess so near, and set them from the nethermost cross-stroke at the beginning of the line of sines forward. And thus do for all under two degrees, be it sine or tangent: but from thence to sine 5 degr. 45 min. or tangent 5 degr. 43 min. (As suppose the sine of 4 degr. whose Logar. beside the characteristick is 843:) you shall take the distance between 8 inches, 4 tenths, 3 cent. and ten inches, and apply that distance from the middle-stroke down-ward: and so of the rest of the quarter. But for all both sines and tangents in this first foot: you may by their Logarithmes strike them with a square, as you did the line of numbers.

Now for the upper-part shewed by the characteristick for all sines and tangents to 20 degr. as suppose the tangent of 20 degr. the Logarithmes of 20 degr. tangent is 56: set it from the middle-stroke forward, but from thence to the sine of 90, and tangent of 45 degr. as the sine of 40, whose Logar. is

808; take the distance between it and the middle-cross-line, and apply it in the line of sines from the upper cross-stroke down-ward: then number all the whole degrees to ten, with 1, 2, 3, and after that in the sines with 20, 30, 40, &c. to 90, and the tangents with 10, 20, to 45, and back with 50, 60, to 80 degrees.

Lastly, for making the sextant of chords.

Set a pair of beam-compasses, with a beam of willow, deal, or sallow, near half an inch thick, and ••/4 broad; make a little nut of good tough wood, with a mortes in it, that the beam may slide in it to and fro, indifferently stiff, and in all places a∣like, with a short prick, or little piece of an aule-blade in one end, and another longer in one edge of the beam hard by the end, so long from the beam as the other point is. If it goeth not stiff enough to stand and tran with at any place; make the mortes a little the deeper one way to put in a wedge, or else help your self with a screw-pin, then go to some smooth loft boards, opening your compasses to 23½ inches, and with that wideness tran an arch, that may be two foot long at the least, and with each foot of the compasses make a prick in the said arch, and set it likewise upon the Rule; then divide that space in the arch into two equal parts, which will be 30 degr. a piece, and each of them into three apiece, which will be 10 degr. apiece, and each of them into two, which will be five apiece, and each of them into five simple ones. Then take them off from the floor, and set them on the Rule, one after another, and number them with 10, 20, 30, 40, 50, 60, and this will be wonderfull beneficial in Dialling, and also in many other things, as to divide a circle into any number of equal parts, or to make an angle of any number of degrees, or to finde the quantity of any angle, and so by the line of foot-measure you may also divide a streight line into as many parts as you will.

Now as I have shewed the use of all the lines on the other side of the Rule, and also of both the out-side lines on this side; so for the other three I must content my self to shew

you the use in general: for if I should descend to particulars, all the paper in Cambridge would be too little to hold them. First therefore, you see already, that as by the line of foot-measure, and Table of Logarithms these lines are made; so may you by these lines finde the Logarithme of any absolute number, tangent or sine, as if it were by the Table of Lo∣garithms.

Secondly, By these two lines of numbers and foot-measure may be resolved all questions whatsoever, that common Arith∣metick can resolve. And more; for hereby may be resolved all questions of Interest, Purchases, Annuities, &c.

Thirdly, By these three lines of numbers, sines, and tangents is resolved the whole doctrine of Triangles, and whatsoever may be performed by them, either in Measuring, Dialling, Geo∣graphy, Geometry, Arithmetick, Navigation, Cosmography, Astronomy, &c.

But, because (gentle Reader) I would have thee learn now to go alone; I will commit these to thine own consideration, knowing that that chicken that will peck up never a corn, but what the hen puts in the mouth, will never be a fat chicken.

Now if the Rule of three is accounted of all men worthy for its excellency of the name of the Golden-Rule (which is but the least part of the use of one of the lines of this Ruler) then justly may this Ruler be called the Golden-Ruler.

FINIS.