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. II. The use of the Dial.

PROB. 1. How to hold the Dial in time of Ob∣servation.

Hang the string of the Dial over your thumb, on your left hand, (or you may hold it between your thumb and the middle of your fore-finger) and stretch your 4 fingers streight out, and let the Dial hang at liberty, just touching the palm of your hand, that it may be steady, then turn your whole body about, till the edge of the Dial (or your fingers ends) be just against the Sun: then shall you see the shadow of any thing stuck in the Cen∣tre, though never so short, to reach quite through the Dial, and then it is held right.

PROB. 2. To find the Suns Altitude.

Stick a pin (the smaller the better) in the Centre, and hold it up as be∣fore, and the shadow will shew on the limb the Suns Altitude required.

Example.

At 8 of the clock on the 11 of June in the morning, I wonld know

the Suns Altitude; I hold it up as before, and I find it to be 36.46, that is 36 Degrees and 46 Minutes, each Degree being 60 Minutes, as in the Tables of Reduction.

PROB. 3. To find the perpendicular height of any thing by its shadow, by the line of shadows.

Hold up the Dial by the thred as before, and look on what division of the line shadows, the shadow of the pins cuts, that is the true height or length of the shadow, by which to get the height of any perpendicular thing, or the very top of any leaning thing, that causeth the shadow.

Example.

On the same 11 of June at neer 9 a Clock in the morning, I hold up my Dial, and I find the shadow to fall just on 1, on the line of shadows, or 45 on the Degrees, therefore I say, that the height of the object, that causeth the shadow, and the shadow are both of one length: but if it had

fallen on 2, (that is to say, of right shadow) then the object is, but half the length of the shadow measuring upon a level ground, from the end of the shadow, to right under the object that causeth the shadow; if it falls on 3, the shadow is 3 times as long as the thing is, and so to 12, to 12 times longer, and the strokes between note one tenth, 2 tenths, 3 tenths, &c. more; but if it falls beyond 1, on con∣trary shadow, then the shadow is shorter accordingly, as will appear very plain with a little practice.

Or rather thus by the Numbers.

Count the middle 1 on the Rule, as 1 at 45 on the Dial, then if the Sun be under 45, count them on the rule toward 10, and if above 45 the contrary; then as the parts cut are to the middle 1, so is the length of the shadow to the altitude required.

PROB. 4. The Use of the Quadrat.

To use the Quadrat, you must have a hole in the other end of the hori∣zontal

line, and also some where in the Dial, square (from the Centre) to the horizontal line; also you must have a thred and plummet, then the use is thus. Stick a pin in the Centre, and thereon hang the plummet, then put a pin in the other whole, that is perpendicular to the horizontal line, and just over the Centre; and hold up the Dial in your right hand, and make the string to play evenly by the Su∣perficies of the Dial, when you see the object right against both the pins; then observe what stroke is cut by the thred on the line of quadrat, (or shadows for that may be used so also,) for if you go backwards or for∣wards till you make the thred to fall on 1, in the shadows or on 50 in the quadrat, then is the height of the House, Steeple, Tree, or the like, equal to the distance, between you and it, adding the height of your eye to it. But if it had fallen on 25 of the quadrat, or 2 on the shadows, then the distance had been twice as much

as the height, (if right shadow) but for contrary shadow, the contrary. I shall say no more to this, only give you a caution, that if you look from the height of any place downwards, then you must put that pin next the Centre to your eye, and look down∣wards to your object, and then the side which before was right shadow, will become contrary shadow, and the contrary. Note one thing further, that if your instrument be a Sextance, or a Circle, and you cannot have all the quadrat, as on a quadrant: you may then move the pin to the hole at the other end of the horizontal line: and you shall see that defect to be supply∣ed. Note lastly, that by heights, we speak only of perpendicular or up∣right heights; and in distances, only of levels, or horizontals.

PROB. 5. How to find unaccessable heights by the quadrat at two Observations.

If the place which is to be measu∣red cannot be approached unto, then

work thus, to find both height and distance, first make choice of a place where looking up I find the thred to fall on 50 in the quadrat, then the di∣stance will be equal to the height. Then make a mark at that Station, and go directly backward in a right line, with the former distance: and make choice of a second Station, where the thred may fall on 25 parts of right shadow, then this second Station is double to the height, and also to the distance, departed from the first Sta∣tion: and the half therefore is the height, and first distance. But if it be so, you cannot come to take such a height as 50 and 25, then take as you may, as suppose one be at 25, and the other at 20, and suppose the height to be 100. I find that, As 25 the parts cut, are to 50 the side of the quadrat, so is 100 the supposed height, unto 200 the distance. And as 20 the second Station, to 50 the side of the quadrat; so is 100 the supposed height, unto 250 the second distance;

wherefore the difference between the Stations should seem to be 50, then if in measuring you find it to be either more or less, then this proportion doth hold as from the supposed diffe∣rence, to the measured difference, so from the supposed height, to the true height, and from the supposed di∣stance, to the true distance.

And now suppose the difference between the two Stations were found to be 30, by measuring, Then as 50 the supposed difference, to 30 the true difference, so is 100 the supposed height, to 60 the true height; And 200 the supposed distance to 120 the true; and 250 at the second Station, unto 150 the distance; the like rea∣son holdeth in all other examples of this kind, and if an Index with sights were fitted to the Centre, it might serve for all other horizontal distances by the same reason.

The Ʋse of the Almanack.

PROB. 6. Having the Day of the Week to find the Day of the Month for ever.

First find what day of the Week the first of January is on; which is thus done: First find the Dominical Let∣ter for the last Leap-year, set down in the Almanack: the next letter is for the next year following, and so till you come to the year you look for: And note, every Leap-year hath two Dominical letters, viz. the next be∣fore it, till the 24 of February, and that over it for the remainder of the year: Having found it, reckon from (A) either backwards or forwards, (always calling (A) Sunday) you shall find what day is the first of Ja∣nuary.

Example.

For the year 1656 (F) is the Do∣minical Letter; therefore say (A) Sunday, (G) Monday, (F) Tues∣day, and that is the first of January;

and then make use of that thus: On the first Tuesday in the beginning of February, I would know the day of the Month? Among the Months look for 12, which is for February, recko∣ning from March, (which is always the first Month) and right under •••• you have 5, for the fifth day, being the first Tuesday in February, and 12, 19, 26 for the other Tuesdays in Februa∣ry: But now for the other Months after March, you must say Wednes∣day, the reason is, because February hath 29 days, and the Leap-year two Dominical Letters, viz. F. and E. then reckon from E to A, and it falls on Wednesday, which use thus in the year 1656, and all other Leap-years: As, in the beginning of August on Thursday, what day of the Month is it? August is the sixth Month, look for 6 among the Months, and right under it you have 6, which is Wed∣nesday, therefore 7 is Thursday, and the first Thursday in August. But now for 1657. I find that Thursday is

the first of January; saying thus, (A) Sunday, (B) Saturday, (C) Friday, (D) Thursday: And so it is all the year long, in all the Months; for having found the Moneth, all the days right under are Thursdays, and then reckon onwards, or backwards for any other of the Week-days, and you have your desire, for any yearpast, present, or to come.

PROB. 7. To find the Epact, and by that the Moons age any day of the Month.

On the Leap-year you have it set down in the Almanack for the next year; add 11. and you have your de∣sire. And for the next year adde 11 to that, and so to the next leap-year: But if by so adding it exceed 30, then take away 30, and the remain is the Epact.

Having the Epact, add to it the day of the Month, and the number of the Month from March also, (in∣cluding both the Moneths) and if they come not to 30, that is the Moons

age; but if they exceed 30, and the Month hath 31 days, then Substract 30, and the remain is the age, but if the month have but 30 days, then substract but 29, and the remainder is the age of the Moon required.

Example.

In July, 1656. on the 20 day the Epact is 14. then 14.20 and 5 added, is 39. from which take 30, rest 9 days old on the 20 of July, 1656. the Moons age sought for.

PROB. 8. To find the hour of the day.

Having found the day of the month by the Almanack, you must find the mark, or the space between two marks in the Kalender, representing that day; which do thus:

Look for the first letter, or name of the month in the Kalender, ac∣cording to the time of the year, then reckon from thence to the day you are in, either by 5, 10, 15, 20, 25, 30, 31, if the parts are so divided, as in small Instruments they cannot well

be more; but if you have single days, every fifth and tenth is known from the rest by a longer stroke, and the last day by the longest stroke: Well, having found the day, or the place between two strokes representing it, lay a thred from the centre over that day, (or for want of a thred, stick a ••in in the centre, and cause the sha∣dow to fall upon the day) and then observe on which, or between which of the 25 or 19 lines the thred cuts the 12 of clock line, for on that line must you look for the hour all that day: Before I come to example, I shall hint a plain word of the reason of this, which I find some to marvel at; The hour of the day in this, and in most Instrumental-Dials, is given by the Suns height; now all men know the Sun is not so high in Win∣ter as in Summer, therefore the Sum∣mer hour lines will not serve the Winter; and also all men know they lengthen by degrees gradually, there∣fore the Winter and Summer 12,

and consequently the rest of the hour lines, run sloping upwards and down∣wards, as the days lengthen or shorten. This being premised and considered, an easier Dial (all things considered) cannot be had. Now for an Example or two:

Having found out the parallel of Declination, (for so is it called) if there be 25 lines, (or of the Suns ri∣sing, if there be but 19) you may ea∣sily know it by the name at the end of it, or by being a prick-line, or the next to, or the 2 next to a prick line, &c. hang or hold the Dial up, as was taught in the 1 Problem, and you shall have the exact hour of the day, among the Summer or Winter hours, accor∣ding to the time of the year.

Example.

On the 2 of Aug. 1656. I look for (A) in the lower line of the months because the days shorten, and laying a string (or causing a shadow to fall) from the centre upon the 2 of Au∣gust, which (if it hath not a particu∣lar

stroke for it) is a little beyond the long stroke by the (A) and toward the (S), and I observe the thred to cut upon the line of Declination, cal∣led 15, and also it is a prick line: (in one of 25 lines (but almost midway between the first, beyond a prick line) and may be called the line of the Suns rising, at 4. and 41 min.) then I hold up my Dial, and find at 8 a clock the shadow to cross the 8 of clock line; just in the prick line, and at the same instant, the Suns altitude is 30.15, and the quadrat is 29, and the line of shadows is 1. and 7 tenths, (that is the shadow of a yard (or any thing) held upright, is the length of the yard, and 7 tenths more of another length or yard) and note, that at 4 a clock the same day the shadow will fall in the same place exactly, as was hinted before; for equal hours from 12. the Sun hath the like altitude at all times of the year; and if it is mor∣ning, the height increases; if after∣noon, then it decreaseth, so that two

observations will resolve the que∣stion. But note,

First for the months of June and Decemb. where the days are close to∣gether, the reason is, because the days at that time lengthen or shorten but a little; so must their spaces be on the instrument; if you should miss 3 or 4. days there, it makes no sensible error, take near as you can, and it sufficeth. Also note the hours of 11 and 12 are neer together, therefore you must be so much the more cauti∣ous in observing to hold the Dial wel, and to look just on, or between the parallel of declination or rising, and at 12 of the clock you may look in the Kalender for the day of the month, for just on that day will the shadow be at 12 of the clock, and short of it (increasing) before, (but decreasing) after 12.

Note also on the 10 of March, and 13 of September, you must ob∣serve in the upper line; but on the 11 of June, and 11 of December on the

lowest line, as the rules rehearsed make manifest.

Lastly, if you meet with a Dial that hath the Kalender of Months on the backside, then it is but laying a thred over the day, and on the line of Declination, the thred cuts the corre∣spondent number of Declination, as before; also the rising, and true place, and amplitude, as I hinted before; Then having the number, look for the line on the other side that shall have the same number, and proceed as be∣fore. Thus much shall suffice for the Dial particular for one latitude.

The use of the other line to make it General, as also of a Joynt-rule to find the hour and azimuth, I shall refer you to the Book of the Joynt-rule, a book of this volume, fit to be bound up with it, being a very useful peice for Dialling, Geometry, Astronomy, and Navigation, and many other Ma∣thematical Conclusions, and a porta∣ble universal Sea-Instrument as any whatsoever extant.