1. To Rectifie or set the Rule to his true Angle.

OPen the Rule to 60 degrees, which is done thus, (indifferent­ly:) make the lines on the head, and the lines on the other leg, meet in a streight line; then is the Scale of Al­titudes and the line of Hours set to an Angle of 60 degrees, the rectifying point, being the center of that Angle; Or to do it more exactly, do thus: put one point of a pair of Compasses into the rectifying point, then open the other to 10, 20, 30, or 40, on the Scale of Altitudes, the Compasses so opened, and the point yet remaining in the re­ctifying point, turn the other to that margenal line in the line of hours, that cuts the rectifying point, and there stay it; then remove the point that was fix­ed [Page 5] in the rectifying point, and open or shut the Rule, till the point of the Compasses will touch 10, 20, 30, or 40, being the point you set the Com­passes too in the Scale of Altitudes, in the innermost line that cuts the center, and the rectifying point, then is it set exactly to 60 degrees, and fitted for observation.

2. To finde the Suns Altitude at any time.

Put a pin in the center hole, at the upper end of the Scale of Altitudes, and on the pin hang a thread and plum­met; then if the Sun be low, that is to say, under 25 degrees high, as in the winter it will always be, then lift up the moveable leg, where the moneths and the degrees be, till the shadow of the end fall just on the meeting of that leg with the head, then the thread shall shew the Suns altitude, counting from 60|0 towards the head, either 10, 20, 25, or any degree between.

But if the Sun be above 25 or 30 de­grees [Page 6] high, lift up the head leg till the shadow of that play as before, or make the shadow of the pin in the center hole play on the innermost line of the Scale of Altitudes where the pin stand­eth, then the thread will fall on the degree, and part of a degree that his true altitude shall be.

But if the Sun be in a cloud, and can not be seen so as to give a shadow, then look up along by the head-leg, or move­able leg, just against the middle of the round body of the Sun, and the thread playing evenly by the degrees, shall show the true altitude required. The like must you do for a Star, or any other object, whose altitude you would find.

3. Having found the Suns altitude, and the day of the moneth, to finde the hour of the day.

Whatsoever you finde the altitude to be, take the same off from the Line of Altitudes, from the center downwards with a pair of Compasses, then lay the thread (being put over the pin) on the [Page 7] day of the moneth, then put one foot of the Compasses in the line of hours, in that line that cuts the rectifying point, and carry it further off, or nigher, till the other foot of the Compass being turned about, will just touch the thred, at the nearest distance, then the point of the Compasses on the line of hours, shall shew the true hour and minute of the day required.

4. To finde the Suns rising any day in the year.

Lay the thread on the day of the month, and in the line of hours it sheweth the true hour and minute of the Suns rising or setting; for the rising, count the morning hours; and for the setting, count the evening hours.

5. To finde if any place lye level, or nor.

Open the rule to his true angle of 60 degrees, then set the moveable leg upon the place you would make level, and if the thread play just on 60 degrees, it is a true level place, or else not.

6. To try if any thing be upright or not.

Hang a thread and plummet on the center, then aply the head leg of the rule to the wall or post, and if it be up­right, the thread will play just on the innermost line of the scale of altitudes, or else not.

1. To finde the Suns Declination.

LAy the Thread on the day of the moneth, then in the line of degrees you have the declination. From March the tenth toward the head is the Decli­nation Northward, the other way is Southward, as by the time of the year is discovered.

3. To finde the Suns right Ascension in hours and minutes.

Lay the thread as before, on the day of the moneth, and in the line of right Ascension you have the hour and mi­nute required, computing right accor­ding to the time of the year, that is, be­gin at the tenth of March, or ♈ Aries, and so reckon forwards and backwards as the moneths go.

4. To finde the Suns Amplitude at rising or setting.

Take the Suns Declination out of the particular Scale of Altitudes, and lay it the same way as the Declination is, [Page 15] from 90 in the Azimuth Scale, and it shall shew the Amplitude from the east or west, counting from 90. Example: May the tenth it is 33. 37.

To finde the Amplitude or Azimuth of the fixed Stars; also their rising, setting, and southing.

1. FIrst for the Amplitude, take the stars declination from the parti­cular [Page 30] Scale of Altitudes, and lay it from 90 in the Azimuth-line, and it shall shew his amplitude from East or West toward South or North, accor­ding to the declination, and time of day, morning or evening. The same work is for the sun.

2. To finde the Azimuth, work as you did for the Sun at the same de­clination the star hath, by Chapter 5. and you shall have your desire.

3 To finde the stars rising and set­ting, lay the thread on the star among the moneths, and in the line of hours it shews the stars rising and setting, as you counted for the Sun; but yet note this is not the time in common hours, but is thus found: Adde the [Page 31] complement of the Suns Ascension, and the stars right Ascension, and the stars hour last found together, and the Sun, if less than 12; or the remain 12 being substracted, shall be the time of his rising in common hours; but for his setting, adde the stars setting last found to the other numbers, and the sum or difference shall be the set­ting.

4. To finde the time of the southing of any star on the Rule, or any other whose right ascension and declination is known, Substract the Suns right ascension from the stars, increased by 24, when you cannot do without, and the remainder, if less than 12, is the time required, in the afternoon or night before 12; but if there remain more than 12, substract 12, and the residue is the time from mid-night to mid-day following.

5. To finde how long any Star will be above the Horizon.

Lay the thread to the star, and in the hour-line it sheweth the ascen­sional difference, counting from 90; then note if the star have North declination, adde that to 6 hours, and the sum is half the time; if south, sub­stract it from 6, and the residue is half the time; and the complement of each to 24 being doubled, is the whole Nocturnal Arch under the Horizon.

To perform the fore-going work in any latitude, as rising, amplitude, ascen­sional difference, latitude, hour, and azimuth, wherein I shall give onely the rule, and leave out the examples for brevity sake.

1. FOr the rising, and setting, and ascensional difference, being all one, do thus: Take the Suns declina­tion out of the general Scale of Alti­tudes, then set one foot of the Com­passes in the colatitude on the same scale, and with the other lay the thred to the nighest distance; then the thred so laid, take the nighest distance from the latitude to the thread, with that distance set one foot in the Suns de­clination, counted from 90 toward the center, and the thread laid to the nearest distance, shall in the degrees [Page 34] shew the ascensional difference re­quired, counting from 90 at the head toward the end of the Rule; and if you reduce those degrees and mi­nutes to time, you have the rising and setting before and after 6, according to the declination and time of the year.

2. To finde the Suns amplitude.

Take the Suns declination, and set­ting one foot in the colatitude, with the other lay the thread to the nearest distance, and on the degrees it shew­eth the Suns amplitude at rising or setting, counting as be [...]ore from 90 to the left end of the Rule.

3. Having amplitude and declina­tion, to finde the latitude.

Take the declination from the gene­ral scale, and set one foot in the am­plitude, the thread laid to the nearest distance in the line of degrees, it sheweth the complement of the lati­tude required, or the converse.

4. Having latitude, Suns declination and altitude, to find the height at 6, and then at any other time of the day and year.

Count the declination in the de­grees from 90 toward the end, there­to lay the thread, the least distance from which to the latitude in the ge­neral Scale, shall be the Suns height at 6 in the summer, or his depression in the winter. The Compasses stan­ding at this distance, take measure on the general Scale of altitudes, from the beginning at the pin towards 90, keeping one point there, open the other to the Suns altitude, thus have you substracted the height at 6, out of the Suns altitude; but in winter you must adde the depression at 6, which is all one at the same declina­tion with his height at 6 in summer, and that is done thus: Put one point of the Compasses so set in the gene­ral Scale to the Suns Altitude, then turn the other outwards toward 90, [Page 36] there keep it, then open the Compas­ses to the beginning of the Scale, then have you added it to the Suns alti­tude; having this distance, set one foot in the colatitude on the general Scale, lay the thread to the nearest distance; the thread so laid, take the nearest distance from 90 to the thred, then set one foot in the declination, counted from 90, and on the degrees it sheweth the hour from 6, reckon­ing from the head, or from 12, coun­ting from the end of the Rule.

I shall make all more plain, by ma­king three Propositions of it, thus:

6. But in winter you must do thus:

By the second Proposition of the [Page 39] ninth Chapter, finde the Suns Am­plitude for that day, then take the altitude from the general Scale of al­titudes, and putting one point in cola­titude, lay the thread to the neerest distance, then the neerest distance from the latitude must be added to the Suns Amplitude; this distance so added must be set from the coalti­tude, and the thread laid to the highest distance, and in the line of de­grees, it gives the Azimuth from south, counting from the end of the rule, or from the East or West, counting from the head or 90 degrees.

1. First a particular for the Substile.

COunt the plains declination on [...]he Azimuth scale, from 90 to­ward the end, and thereunto lay the thread, in the line of degrees it shews the distance of the substile from 12.

2. For the height of the stile above the substile.

Take the Plaines Declination from 90 in the Azimuth line, but counted from the South end, between your compasses: and measure it in the particular scale of altitudes, and [Page 41] it shall give the height of the stile re­quired.

3. For the inclination of Meridians:

Count the substile on the particu­ler scale of Altitudes, and take that distance between your compasses, measure this distance on the Azimuth line from 90 toward the end, and counting that way it sheweth the in­clination of Meridians required.

4. To finde the Angle of 6 from 12.

Take the plaines declination from the particular scale of altitudes, and lay it from 90 on the Azimuth scale, and to the Compasses point lay the thread: then on the line of de­grees you have the complement of 6 from 12, counting from 60 toward the end.

[Page 42] Note this Rule (as this line is drawn) doth not give this Angle ex­actly, neither will it be worth the while to delineate another line for this purpose. But if it be re­quired it may be done, but I rather prefer this help, the greatest error is about the space of 45 minutes of the first degree in the particular scale of altitudes; so that if you conceive those 45 minutes to be divided as the particular scale of altitudes is, like a natural sine, and if your declina­tion be 30, then take half the space of the 45 minutes less, and that shall be the true distance to lay on the Azimuth line from 90 whereunto to lay the thread.

5. To perform the same generally by the general scale of altitudes; and first for the stile.

[Page 43] Lay the thred to the complement of the latitude, counted in the degrees from the head toward the end, then the nightest distance from the comple­ment of the plaines declination to the thread, taken and measured on the general scale, from the center, shall be the stiles height required.

6. To finde the inclination of Meridians

Take the plaines declination, from the general scale, and fit it in the complement of the stiles elevation, and lay the thread to the neerest di­stance, and on the degrees it sheweth the inclination of Meridians required.

7. For the substile,

Count in the inclination of Meri­dians on the degrees from 90, and thereto lay the thread, then take the least distance from the latitudes com­plement to the thread, set one foot of that distance in 90, and lay the thread to the neerest distance, and in the [Page 44] degrees it shall shew the substile from 12 required.

8. For the angle of 6 from 12.

Take the side of the square, or the measure of the parallel from 12, and fit it in the cosine of the latitude, and lay the thread to the nighest distance, then take out the nearest distance from the sine of the latitude to the thread, then fit that over in the sine of 90, and to the nearest distance lay the thread, then take the nearest di­stance from the sine of the plains de­clination to the thread, and it shall reach on the parallel line, or side of the square, from the Horizon to 6 a clock line required.

And first by the needle because the easiest.

For this purpose you must have a needle well touched with the Load­stone of about three or four inches long, and fitted into a box somewhat broader then one of the legs of the sector, with a lid to open and shut; and on the inside of the lid may be drawn a South erect Dial, and a wire to set the lid upright, and a thread to be the Gnomon or stile to that Dial: it will not be a miss also to ex­tend the lines on the Horizontal part for the same thread is a stile for that also. Also on the bottom let there be a rabbit, or grove, made to fit the leg of the rule or sector; so as being pres­sed into it, it may not fall off from the rule, if your hand should shake, or you cease to hold it there. This being so fitted, the uses follow in their or­der.

Put your box and needle on that leg of the rule, that will be most fit [Page 54] for your purpose, and also the north end of the needle toward the wall, if it be a south wall; and the contrary, if a north, as the playing of the needle will direct you, better then the way how in a thousand words, then open or close the Rule, till the needle play right over the north and south-line, in the bottom of the Box: then the complement of the Angle that the Sector standeth at (which may always be under 90 degrees) is the declina­tion of the Plain. But if it happen to stand at any Angle above 90, then the quantity thereof above 90, is the de­clination of the wall.

To finde the quantity of the Angle the Sector stands at may be done two ways: first by protraction, by laying down the Rule so set on a board, and draw two lines by the legs of the Se­ctor, and finde the Angle by a line of Chords. Secondly, more speedily and artificially, thus: By the lines of Sines being drawn to 2. 4, or 6 de­grees asunder. The Sector so set, take [Page 55] the parallel sine of 30 and 30, and measure it on the lateral sines from the center, and it shall reach to the sine of half the Angle the line of sines stands at, being more by 2, 4, or 6 degrees then the sector stands at, be­cause it is drawn one, two, or three degrees from the inside.

Or else take the latteral sine of 30 from the center, and keeping one foot fixed in 30, turn the other till it cross the line of sines on that line next the inside, and counting from 90, it shall touch at the Angle the line of sines standsat, being two degrees more then the Sector stands at; the lines being drawn so, will be (as I conceive) most convenien.

To finde a Declination by the Sun.

First open the Rule to an Angle of 60 degrees, as you do to finde the [Page 58] hour of the day, and put a pin in the hole, and hang the thread and plum­met on the pin; also you must have another thread somewhat longer and grosser then that for the hour in a readiness for your use.

Then apply the head leg to the wall, if the sun be coming on the Plain, and hold the Rule horizontal or level, then hold up the long thread till the shadow falls right over the pin, or the center hole, at the same instant the shadow shall shew on the degrees, how much the sun wants of coming to be just against the Plain, which I call the Meridian or Pole of the Plain, which number you must write down thus, as suppose it fell on 40, write down 40-00 want: then as soon as may be take the Suns true altitude, and write that down also, with which you must finde the Suns Azimuth▪ then substract the greater out of the less, and the remainder is the declination required. But for a general rule, take this: if the Sun do [Page 59] want of coming to the Meridian of the place, and also want of coming to the Meridian of the Plain, then you must alway substract the greater number out of the less▪ whether it be forenoon or after-noon, so likewise when the Sun is past the Meridian, and past the Plain also. But if the signes be unlike, that is to say, one past the Plain, or Meridian, and the other want either of the Plain, or Meridian, then you must add them to­gether, and the sum is the declination from the South. Which rule for bet­ter tenaciousness sake take in this homely rime.

The further illustration by two or three Examples.

Suppose on the first of May, in the forenoon, I come and apply the Rule, being opened to his Angle of 60 de­grees to the wall, (viz. the head leg, [Page 60] or the leg where the center is) and holding up my thickest thread and plummet, so as the shadow of it cros­seth the center, and at the same in­stant also on 60 degrees, then I say the sun wants 60 deg. of coming to the Meridian of the Plain; at the same instant, or as soon as possible I can, I take the suns altitude, as before is shewed, and set that down, which suppose it to be 20 degrees, then by the rules before, get the Suns azimuth for that day, and altitude; which in our example will be found to be 94 degrees from the (south or) Meridian, then in regard the signs are both a­like, i. e. want, if you substract one out of the other, there remains 34 the declination required; but for the right denomination which way, either north or south, toward either east or west, observe this plain rule: First, if the Sun come to the Meridian or Pole of the Plain, before it come to the Meridian or Pole of the place, then it is always an East-plain; but if [Page 61] the contrary, it is a West-plain, that is to say, if the Sun come to the Me­ridian or Pole of the place, before it comes to the Meridian or Pole of the Plain, then it is a West-plain. Also if the sum or remainder, after addi­tion or substraction, be under 90, it is a South-plain; but if it be above 90, it is a North-plain. Also note, that when the sum or remainder is above 90, then the complement to 180, is always the declination from the north toward either east or west; So that according to these rules in our exam­ple it is 34 degrees South-east.

Again in the morning, Iune 13. I apply my rule to the wall, and I finde the Sun is past the Pole or Meridian of the Plain 10 degrees, and the al­titude at the same time 15 degrees, the Azimuth at that altitude, and day in this latitude, will be found to be 109 degrees want of south or pole of the place; therefore unlike signs, and to be added, and they make up 119 degrees, whose complement to 180 [Page 62] is 61; for 61 and 119 added, make up 180, therefore this Plain declineth 61 degrees from the north toward the east.

Again the same day in the after­noon, I finde the Azimuth past the south or meridian of the place 30 de­grees, and at the same time the Sun wants in coming to the meridian or pole of the Plain 10 degrees, here by addition I finde the declination to be 40 degrees south-west.

Note what I have said in these three examples, is general at all times; but if it be a fair day, and time and op­portunity serve, to come either just at 12 a clock, when the Sun is the me­ridian or pole of the place; or just when the Sun is in the meridian or pole of the Plain, then your work is onely thus:

First if you come to observe at 12, then applying your rule to the wall, and holding up the thread and plum­met, how much so ever the Sun wants or is past the pole of the Plain, that [Page 63] is the declination, if it be past it is east-wards; if it wants, it is south-west-wards; if neither, a just South Plain, and then the poles, or Meridians of place and Plain, are the same.

But secondly, if you come when the Sun is just in the pole of the Plain, then whatsoever you finde the Suns Azimuth to be, that is the decli­nation; if it wants of south, it declineth East-wards; if it be past, it declineth West-wards.

Thus I have copiously, (and yet very briefly) shewed you the most ar­tificial way of getting the declination of any wall, howsoever situated.

Note if the Sun be above 15 de­grees wanting of the Meridian of the Plain, your rule will prove defective in taking the Plains Meridian when the center leg is next the wall, then you must turn the other leg to the wall, and then you finde a supply for all an­gles to 45 degrees past the Plain. But for the supply of the rest which is 45 degrees, do thus: open the rule till the [Page 64] great line of tangents & the outside of the leg make a right angle, for which on the head you may make a mark for the ready setting, then making the inside of the leg at the end of 45, as a center, the tangents on the other leg supply very largely the defect of the other­sides. Or if you set on the box and needle on the rule, and open or shut the rule till the shadow of the thread shew just 12, then the Angle the Se­ctor stands at, is the complement of what the sum wants, or is past the meridian or pole of the Plain.

Another way to perform this Geometri­cally for all erect Dyals with centers.

When you have drawn a line of 12, and appointed a center, make a Geo­metrical square on that side 12, as the stile must stand on, as A B C D, the perpendicular side of which square may also be the parallel as before.

Then first, fit the side of the square in the sine of the latitude, and take out the cosine of the latitude, and fit that in the sine of 90; the Sector so standing, take out the sine of the de­clination, and lay it from B toward C, unto [...], and draw A E for the substile, then upon the point E in the substile, raise a perpendicular, and take out the sine complement of the Plains declination, and set it on that perpendicular from E to F, for the [Page] [Page]

[Page 67] height of the stile above the Plain.

Again, fit the side of the square in the cosine of the latitude, and take out the sine of the latitude, and fit that over in the sine of 90, then take out the [...] sine of the declination, and lay it from D to G for the hour of 6, and draw the line A G for the 6 a clock hour line.

Then again, fit the side of the square (or the distance of the parallel in the other way, when you want a secant, or your secant too little) in the sine of 90, and take out the cosine of the de­clination, fit that in the cosine of the latitude, then take out the sine of 90, and lay it from the center on the line of 12, and [...] 6 in the side of the square, and by those two points draw the contingent line▪ and then fit those points or distances in the tangent of 45, and lay down the hours as in the former part of this chapter; but if you want the hours before 9 in west-decliners, or the hours after 3 in east-decliners, and the 6 fall too high a­bove [Page 68] the horizontal line on that side, you may supply this defect thus:

Take the measure with your Com­passes from 3 to 4 and 5, upwards in the west-decliners, or from 9 to 7 and 8 in east-decliners, and lay it upwards from 9 or 3 on the deficient side, in the parallel as it should have been from 6 downward in south or upward in north Plains, and you shall see all your defective points to be compleat­ly supplied, whereunto draw the hour­lines accordingly.

An Advertisement relating to Dialling.

These directions are sufficient for any Horizontal, direct, and declining vertical Dials, but for all other, as East or West inclining and reclining, direct, or reclining and inclining, and declining polars; and all decli­ning and inclining: and reclinimg Plains, the perfect knowledge of their affections and scituations is very hard to conceive, but much more hard to remember by roat, or knowingly.

Therefore for the better help of the imagination to conceive aright of their affections, and setuations you may be furnished with an Instrument in brass, at the house of Mr. Walter Hayes, neer the Popes-head Tavern in Morefields, at the Sign of the Cross-daggers, or at the house of Iohn Brown, at the Sign of the Sun-Dial in the Minories, or at Mr. Henry Suttons in [Page]

[Page] [Page 77] Thredneedle-street behind the Ex­change. Which Instrument being righ [...]ly made, you may rectifie to any latitude, and to any declination or reclination, and being so set, it will give the substile, stile, inclination of Meridians, and every hours Ark on the Plain by occular inspection to half a degree, and shew you all the affections straightway: howsoever situated, as which Pole is elevated, and which way applied, and what hours will be useful, &c.

A representation of which instru­ment take in this place, let the circle E S W N, represent the Horizon S AE P N the Meridian, AE W AE E the Equinoctial, P C P the Pole, being the points of an Axes made of th [...]ead, and passing through the center C W C E being shaddowed, repre­sents the Plain, set to a declination, and to a reclination 45 degrees, F the foot that doth support it; moreover the Plain hat [...] an Arch fitted to it, to get the stiles height, and to set it [Page 78] to any angle; also the Horizon and Plain is made to turn round, which could not well be expressed in this figure, or representation of the in­strument: a more liveley and easie help cannot likely be invented, and for cheapness the like may be in a half circle, or made of Past-board.

The use and application of which figure, I shall not now speak of, part­ly because of the facillity in the using thereof, and partly because of the dif­ficulty in description thereof; and lastly, wheresoever you shall buy the same, at either of the three places, the use will be taught you gratis.

Also note, that half of the Instru­ment may be made either in b [...]ass, or pastboard, and be made to fold down in a book, and perform the uses there­of indifferently well, for many pur­poses, as to the Affections of Dials.

Then having discovered the af­fections, you may by the Canno [...]s in the seventh Chapter of Mr. Windgates R [...]le of proportion, finde all the re­quisites, [Page 79] and then to speed the laying down the hours, you may do it by help of the tangent of 45, as you did in erect decliners; for after you have drawn the hours of 6, and three, or nine, and made a parallelogram by two lines parallel to 12, and one pa­rallel to six, then making the distance from six to nine, or three in the line parallel to 12, a tangent of 45, and the distance from 12 to 9, or 3 in the parallel to 6, likewise a tangent of 45, in the sector, and laying off 15, 30. 45, or the respective quarters from 6, and 12, in those two lines, and they shall be the true points to draw the hour lines, by laying a ruler to them, and the center of the Dial; but for those that have no centers, the rule of augmenting the stile in far decliners, serveth for these also.

To perform the same by the line of sines, drawn from the center on the flat-side, and the line of lines, or equal parts or inches in ten parts.

To work these or any other questi­ons by the line of natural Sines and Tangents, on the flat-side, drawn from the center, it is but changing the terms, thus: As the measured distance taken out of the line of lines, or any scale of equal parts, is to the sine of the angle, opposite to that measured side, fitted across from one leg to the other, the Sector so standing, take out the parallel sine of the angle opposite to the enquired side, and that measure shall reach on the line of equal parts, to the measure of the Altitude requi­ [...] ▪

The use of certain lines for the mensu­ration of superficial and solid bodies, usually inserted on Ioynt-Rules for the use of Work-men, of several sorts and kindes.

FIrst the most general and received line for mensuration of Magni­tudes, is a foot divided into 12 inch­es, and those inches into 8, 10, 12, or more parts; but this being not so apt for application to the numbers, I shall not insist of it here, but rather refer you to the Carpenters Rule; yet nevertheless those inches, laid by a line of foot measure, doth by occular inspection onely, serve to reduce foot measure to inches, and inches likewise to foot measure, and some other con­clusions also.

[Page 88] 1. As first, The price of any com­modity at five score to the hundred, either tale or weight, being given, to finde the price of one in number, or one pound in weight.

As suppose at two pence half-peny a pound, (or one) I demand to what cometh the hundred weight, (or five-score,) counting so many pound to the hundred weight?

If you look for two inches and a half, representing two pence two far­things, right against it on the foot measure, you have 21 very near; for if you conceive the space between 20 and 21, to be parted into 12 parts, this will be found to contain ten of them, for the odde ten pence. But for the more certain computation of the odde pence, look how many far­things there is in the price of one pound, twice so many shillings, and once so many pence is the remain­der, which if it be above 12, the 12 or 12s, being substracted, the remain­der is the precise number of pence, [Page 89] above the shillings there expressed; and on the contrary, at any price the C hundred, or 5 score, to finde the price of one, or 1l. As suppose at 40s. the C. or 5 score, look for 40 in the foot measure, and right against it in the inches, you have 4 inches, 3 quarters, and 1/4 of a quarter, which in this way of account is 4 pence 3 far­things, and about a quarter of one far­thing. Thus by the lines, as they are divided, it proceeds to 12 pence a pound; but if you conceive the inch­es to be doubled, and the foot mea­sure also, you shall have it to 24 d. or 48 d. the pound, or one in tale, of any commodity. As at 18 d. a piece, or pound, the price comes to 7l. 10s. the C. for then every ten strokes is 20s in the foot measure, and every inch is 2 pence, and every eighth one farthing.

2. Secondly, for the buying of Tim­ber at 50 foot to a Load, at any price the load, how much a foot.

Here in resolving this, the inches [Page 90] are to be doubled, and the foot mea­sure taken as it is: As at 40 shillings the Load, 40 in the foot measure stands right against 4 inches 3 quar­ters and better, which being doubled, is 9d. 2 far. 1/2 far. near, for the price of one foot; and on the contrary, at 5d. a foot, is 41s. 8 d. a load, &c.

3. For the great Hundred of 112l. to the Hundred, let the space of 12 inches be divided into 112 parts, then the like rule holds for that also.

For the inches being divided into quarters, every quarter is a farthing, and every eighth half a farthing, and every division of the 112 is a shilling, and every alteration of a farthing in the price of a pound, makes a groat in the Hundred, as thus: At 3 pence a pound is 28s. the C. At 3d. 1 q. a pound, 30s. and 4d. the C. At 3d. 1/2 the pound, 32s. 8 d. the C. At 3 d. 3 farthings, 35s. the C. Thus you see that every fraction at a farthing advance, is 4 pence in the Hundred; but for any other account, as 3 pence [Page 91] farthing half farthing, then count the fraction, as 1, 12th part of a shilling, and nearer you cannot come by a bare occular inspection; but the price of the Hundred being given, the price of the pound you have as near by this occular inspection, as any usual Coin is reducable, viz. to the 32 part of a peny, or nigher if you please.

Again note here also, you may dou­ble, or quadruple the price: as to 24d. or 48d. the pound, or any price between. As for example: At 13d. a pound, is 6l. 1s. 4d. the C. At 32d. or 2s. 8d. the pound, is 14l. 19s. the C. and the like by dupling and quadrupling the inches, and the 112 parts, that layeth by it.

4. These lines of equal parts serve as Scales, for the protracting of any Draught of house or field, or the like; also for addition or substraction of any small number.

5. Note that the line of foot mea­sure, may be applied for the reducing of any odde fraction to a decimal fra­ction, [Page 92] as you may fee it in page 64. of Mr. Windgats Arithmetick made easie.

2. The use of the lines of decimal Tim­ber and Board measure.

The lines of decimal Timber and Board measure, are fitted to agree with the tenths, or foot measure, as those lines in the first chapter of the Carpenters Rule, are fitted to the in­ches, and the use of them is thus: And first for the decimal Board mea­sure. Suppose a Board is 1 foot 50 broad, I look for 150 on that line, and from that place to the end of the Rule forwards, toward 100, so much in length must you have to make a foot of superficial or board measure.

2. Or else thus: If you apply the end of the Rule next 100, to one edge of the breadth of a board, or glass, then right against the other edge of the board, on that line of decimal board measure, you shall finde the 10ths and 100s, (or feet 10ths & 100) [Page 93] parts of a foot, that you must have in length to make a foot superficial at that breadth.

3. The use of decimal Timber mea­sure.

The use of this is much like the Board measure, onely here you must have a respect to the squareness of the piece, and not to the breadth onely; for after you know how much the piece of timber or stone is square, in feet and 100 parts, then look that number on the line of decimal Tim­ber measure, and from thence to the end of the Rule, is the length that goes to make a foot of timber.

5. The use of the line of decimal yard measure, also running yard measure, ac­cording to the inches or decimal parts of a foot.

The decimal yard measure, is no­thing else but a yard or 3 foot divided into a 100 parts, and used in the same manner as the foot measure is, for if you take the length, and the breadth in that measure, and multiply it toge­ther, you shall have the content, in yards and 100 parts of a yard.

7. The use of the line of decimal round measure, commonly called Girt-mea­sure, which is when the circumference of a round Cillender, or piller given in inches or ten parts of a foot.

First (for Girt-measure according to inches, being the most usual measure) now much the pillar is about, then look for the same number on the line of Girt-measure, and from thence to the end of the rule, is the length that goeth to make a foot of Timber.

But if it be under 30 inches about, then you must have above two foot in length, and then a table at the end of the line, or a repetition in another [Page 97] line, will supply the defect. But if the line of Girt-measure be divided according to foot measure, then use it as before, seeking the decimal part on the line, and from thence to the end is a foot.

8. The use of a line of solid measure, by having the Diameter of a round piece given in inches, or foot measure.

Take the diameter with a rule, or a pair of Callipers, and learn the mea­sure either in inches, or foot measure according as your line of Diameter is divided. Then look for the same number on the line of Diameter, and from thence to the end of the rule forward, is the length that makes a foot of timber at that diameter (or measure cross the end of the round piece of Timber or stone.)

The Tables of all the under mea­sure for all these lines follow.

[Page 98]

[Page 107] If you would have this Table for 1/2 a brick, take the half of the table for one brick. If for two bricks then double it. If for two and a 1/2 then ad both these together; if for three, double that for one brick and 1/2.

If you have any number of feet of brick work, at half a brick, one brick, or two bricks, or more, and you would reduce it to one brick and half, then say by the line of numbers as 1. 2. 4. 5 or 6 is to three, so is the number of feet at 1/2 1. 2. 2 1/2 or three bricks to the number of feet at one and 1/2.

The use of four scales, called Circum­fence, Diameter, Square equal, Square inscribed.

Suppose you have a circle whose diameter is 10 inches, or 10 feet: and to this circle you would finde the Circumference, or the side of a square equal, or inscribed, or having any one of the three, to finde the other three, do thus:

Take the measure of the Circum­ference, [Page 108] Diameter, or either of the squares, which is first given, and open the compasses to the number of the given measure, in its respective scale: the compasses so set, if you apply it in the scale whose number you would know, you shall have your desire.

The use of Mr. Whites rule, for the measuring of Timber and Board, either by inches or foot measure.

1. ANd first for superficial or board measure, by the inches, the breadth and length being given in inches, and feet and parts, slide or set 12 on any one side, to the breadth in inches or parts on the other side, then just against the length found on the first side, where 12 was on the second side you shall have the con­tent in feet, and 10ths, or 100 parts required. Which by the rules of re­duction by the foot measure, you may reduce to inches and 8 parts.

2. The breadth being given in inches, to finde how many inches in length goes to make a foot of board or flat measure.

Set 12 on the first side, to the breadth in inches on the second side, then look for 12 on the second side, and right against it on the first side, is the number of inches, that goes to make a foot Superficial, at that breadth.

3. To work multiplication on the s [...]iding, or Whites rule.

Set one on the first side, to the mul­tiplicator on the second side, then seek the multiplicand on the first, and right against it on the second, you shall finde the product.

4. To work division on the same rule.

Set the divisor on the first side to one on the second, then the dividend on the first, shall on the second shew the quotient required.

3. To work the rule of 3 direct.

Set the first term of the question, sought out on the first line, to the se­cond term on the other, (or second line:) then the third term sought on the first line, right against it on the second, you shall finde the fourth pro­portional term required.

To work the rule of 3 reverse.

4. Set the first term sought out on the first line, to the second being of the same denomination or kind to the se­cond line, or side. Then seek the third term on the second side, and on the first you shall have the answer re­quired.

6. To work the double rule of 3 direct.

This is done by two workings: As thus for Example.

If 112 l. or 1 C. weight, cost 12 pence the carraiage for 20 miles, what shall 6 C, cost, 100 miles? Say first by the third rule last mentioned, as 1 C. weight to 12, so is 6 C. weight to 72. pence, secondly say if 6 C. cost 72 pence or rather [Page 114] 6s, for 20 miles? what shall 100 miles require? the answer is 30 s. for if you set 20 against 6 then right against 100 is 30, the answer required.

The use of Mr. Whites rule in measu­ring Timber round, or square, the square or girt being given in inches, and the length in feet and inches.

1. The inches that a piece of Timber is square, being given: to finde how much in length makes a foot of Tim­ber, look the number of inches square on that side of the Timber line, which is numbred with single figures from 1 to 12, and set it just against 100 on the other or second side, then right against 12 at the lower, (or some times the upper) end, on the first line, in the second you have the number of feet and inches required.

Example. At 4 1/2 inches square, you must have 7 foot 1 inch 1/3 to make a foot of Timber. But if it be above 12 inches square, then use the sixth Pro­blem of the 5th chapter of the Car­penters [Page 115] Rule, with the double figured side and Compasses.

2. But if it be a round smooth stick, of above 12 inches about, and to it you would know how much in length makes a true foot, then do thus. Set the one at the beginning of the dou­ble figured side, next your left hand, to the feet and inches about, counted in the other side, numbred with single figures from 1 to 12, then right against three foot six 1/2 inches, in the single figures side next the right hand, you have in the first side the number of feet, and inches required.

Example. A piece of 12 inches about, requires 11 f 7 in: fere to make a foot. Again a piece of 15 inches about, must have 8 foot 1/2 an inch in length, to make a foot of timber.

3. But if you would have it to be equal to the square, made by the 4th part of a line girt about the piece, then instead of three foot 6 1/2 inches make use of four foot, and you shall have your desire.

[Page 116] 4. The side of a square being given in inches, and the length in feet, to find the content of a piece of timber.

If it be under 12 inches square then work thus: set 12 at the beginning or end of the right hand side, to the length counted on the other side, then right against the inches square on the right side is the content on the left side

Example. At 30 foot long, & 9 in­ches square, you shall find 16 foot 11 inches for the working this question, 12 at the end must be used.

But if it be above 12 inches square, then ser one at the beginning, or 10 at the end of the right hand side, to the length counted on the other side, then the number of inches or rather feet and inches, counted on the first side, shall shew on the second the feet and parts required.

Example. At 1 f. 6 inch. square, and 30 foot long, you shall finde 67 feet and about a 1/2.

5. To measure a round piece by having the length, and the number of [Page 117] inches about, being a smooth piece, and to measure true, and just measure, then proceed thus:

Set 3 f. 6 1/2 inches on the right side, to the length on the other side, then the feet and inches about, on the first side, shall shew on the second or left, the content required. As at 20 inches about, and 20 foot long, the content will be found to be about 4 foot 5 inches.

But if you give the usual allowance, that is made by dupling the string 4 times, that girts the piece: then you must set 4 foot on the right side, to the length on the other, then at 1 foot 8 inches about, the last example you shall finde but three foot 6 inches.

6. [...] astly, if the rule be made fit for foot measure, onely then the point of 12 is altogether neglected, and one onely made use of as a standing num­ber: and the point at three foot 6 1/2 will be at three foot 54 parts; and the four will be the same, and the same [Page 118] directions in every respect, serve the turn. And because I call it Mr. Whites rule, being the contriver thereof, ac­cording to feet and inches, I have therefore fitted these directions ac­cordingly, and there are sufficient to the ingenious practitioner.

The uses of the lines follow.

As to the use of these lines, I shall in this place say but little, and that for two reasons. First, because this instrument is so contrived, that the use is sooner learned then any other, I speak as to the manner, and way of using it, because by means of first, se­cond, and third radiusses, in sines and Tangents, the work is always right on, one way or other, according to the Canon whatsoever it be, in any book that treats of the Logarithms, as Gunter, Wells, Oughtred, Norwood, or others, as in Oughtred from page 62 to 107.

Secondly, and more especially, be­cause the more accurate, and large handling thèreof is more then promi­sed, if not already performed by more abler pens, and a large manuscript thereof by my Sires meanes, provi­ded many years ago, though to this day not extant in print; so for his sake I claiming my intrest therein, make bold to present you with these [Page 138] few lines, in order to the use of them: And first no [...]e.

1. Which soever of the two legs is set to the first term in the question, that I call the first leg always, and the other being set to the second term, I call the second leg.

2. Secondly, if one be the first or second term, then for the better set­ting the index exactly, you may set it to 100, for the error is like to be the least neerest the circumference.

3. Thirdly, be sure you keep a true account of the number of turnes be­tween the first and second term.

4. Fourthly, observe which way you move, from the first to the second term. To keep the like from the third to the fourth, exept in the back rule of three, and in such cases as the Ca­non requires the contrary.

5. Fifthly in multiplication, one is always the first term, and the multi­plycator or multiplycand the second, and the product always the fourth. Also note that in multiplycation the [Page 139] product of two numbers multiplyed, shall be in as many places as both the multiplycator, and multiplycand, except the least of them, be less then the two first figures of the product; moreover, for your more certain as­singing of the two last figures of four or six, which is as many as you can see on this instrument, multiply the two last in your minde, and the product shall be the figure, as in page 28 of the Carpenters Rule.

6. In division, the multiplicator is always the first term, and one the se­cond, the dividend the third, and the quotient the fourth; also the quotient shall have as many figures as the di­vidend hath more then the divisor, except the first figures of the divisor be greater then the dividends, then it shall have one less. Also note, the fraction after division, is a decimal fraction, and to be reduced as before.

7. Note carefully whether the fourth proportional ought to be a greater or a less, and resolve accor­dingly, [Page 140] and note if one cometh be­tween the third and fourth term, then must the fourth be raised a Radius or a figure more, and be careful to set the hairs exactly over the part repre­senting the number or minutes of any degree.

8. Always in direct proportion, and Astronomical calculation, set the first leg to the first term, and the se­cond leg to the second term, and note how many circles is between, then set the first leg to the third term, and right under the second leg, the same way, and so many turnes between the third and fourth, is always the fourth term required.

The Latitude and Declination given, to finde the Suns height at 6 a clock, Dec. 23. 31. Lat. 51. 32.

Lay the left leg on the Sine of 90, and the other to the Sine of 23. 32, and you shall finde one turn between upwards, then the first leg laid on the Sine of the latitude 51, 32, the o­ther leg shall shew the Sine of 18. 3. (in the second circle above 51, 32.) for the Suns height at 6 required, and this is fixed for one day.

Then in summer time, or north de­clinations, by help of the line of na­tural sines, in the second line, finde the difference between the Suns pre­sent altitude, and the latitude at 6, but in Winter or Southern (signs or) declinations, add the two altitudes to­gether, in this manner. Lay one leg of the index to the natural sine of the altitude at 6, and the other to the altitude proposed, the two legs so set, bring one of them (viz the right) to the beginning of the line of natural [Page 145] sines, and the other shall stay at the difference required, but in Winter set one leg to the beginning of the sines, and open the other to the height at 6, or rather depression under the Horizon at 6, (which is all one at like declinations, North and South) then set the first leg to the present al­titude of the Sun, and the other shall shew the Sine of the sum of both ad­ded together; which sum or differ­ence is thus to be used:

Lay the left leg to the Cosine of the declination, and the other to the secant of the latitude, counted be­yond 90, as far as the secant of 9, 40; or rather lay the left leg on the Co­sine of the latitude, and the other to the secant of the suns declination, then the first leg laid on the sine of the sum in winter, or difference in summer, shall cause the other leg to fall on the sine of the hour from 6, toward noon in winter and summer, except the altitude in summer be less then the altitude at 6, then it is the [Page 146] hour from 6, toward mid-night.

Two other Canons to finde the hour of the day, and Azimuth of the Sun, by one operation, by help of the na­tural sines: and first for the hour.

Having the latitude, the Suns Meri­dian, and present altitude, and decli­nation, to finde the hour from noon.

First lay one leg to the Meridian altitude, in the line of natural sines, and the other to the sine of the alti­tude in the same line, then bring the right leg to the beginning of the line of sines, and the other [...] difference, which difference [...] keep.

Then lay one leg on the [...] the declination, and the other [...] [Page 150] secant of the latitude, and note the turnes between, or rather lay the first leg to the cosine of the latitude, and the other to the secant of the decli­nation, then the legs being so set, bring the first or left leg to the sine of the difference first found, and the other leg shall shew the versed sine of the hour from noon, if the ver­sed sines had been set thus, i. e. the versed sine of 90, against the sine of 90, as in some instruments it is: But to remedy this defect, do thus: keep the right leg there, and open the o­ther to the versed sine of 0. or sine of 90, and note the turnes between, then lay the leg that was on the sine of 90, (or versed sine of 0,) to the versed sine of 90, and the other leg shall shew the versed sine of the hour from noon, counting from 90.

Having the latitude, declination, and Suns altitude, to finde the Azimuth from South or North.

First add the complements of the coaltitude and cola [...]itude together, then if the sum be under 90, take the distance between the cosine of it, and the sine of the declination, in the line [Page 152] of natural sines, and measure it in the line of sines from the beginning, and it shall give the sine of the difference; but if the sum exceed 90, then when the latitude and declination is alike, add the excess to the declination; but if contrary substract one out of other, and measuring the sum or remainder from the beginning of the sines, you have the difference which you must keep.

Then lay the first or left leg to the cosine of the altitude, and the second to the secant of the latitude, or else lay the first to the cosine of the lati­tude, and the other to the secant of the altitude, and more the turnes be­tween, then lay the first leg to the sine of the difference before found, and the other shall shew the versed sine of the Azimuth from noon re­quired, if the versed sines be set as before is expressed, that is to say 90 of right sines, and versed sines toge­ther, and numbred forwards as the sines be: but in the use of this instru­ment, [Page 153] the remedy aforesaid supplyeth the defect.

As the Radius or sine of 90, to the co­sine of the latitude 51. 32: so is the sine of the Suns distance from 6 (in hours and minutes, being turned into degrees and minutes 30, for 8. or 4,)

[Page 155] To the sine of the altitude of the Sun at the time required 18. 07, But for all other times say,

As the sine of 90,
To the cotangent of the latitude 38. 28.
So is the sine of the Suns distance from 6, 30, 0,
To the tangent of the 4 arke 21 40.

Which fourth arke must be taken out of the Suns distance from the Pole 66, 21, (in Cancer) leaveth a resi­due 44 48, which is called a fift ark. But for the hours before and after 6, you must add the fourth arke to the Suns distance from the Pole, and the Sum is the fifth ark. Then say,

As the cosine of the fourth ark 78, 20,
Is to the sine of the latitude 51, 32:
So is the cosine of the residue 45, 12,
To the sine of the Suns altitude at 8, 36, 42, at that declination.

The Use of the Almanack.

Having the year, to finde the day of the week the first of March is on in that year, and Dominical letter also.

First if it be a Leap-year, then look for it in the row of Leap-year, and in the column of week-days, right over it is the day required, and in the row of dominical letters is the Sunday let­ters also: but note the Dominical let­ter changeth the first of Ianuary, but the week day the first of March, so also doth the Epact.

Example. In the year 1660, right over 60 which stands for 1660, there is G for the Dominical or Sunday let­ter, beginning at Ianuary, and T for thursday the day of the week the first of March is on, and 28 underneath for the epact that year, but in the year [Page 166] 1661. being the next after 1660 the Leap-year, count onwards toward your right hand, and when you come to the last column, begin again at the right hand, and so count forwards till you come to the next Leap-year, ac­cording to this account for 61, T is the dominical letter, and Friday is the first of March.

But to finde the Epact, count how many years it is since the last Leap-year, which can be but three, for every 4th is a Leap-year, and adde so many times 11 to the epact in the Leap-year last past, and the sum, if under 30, is the Epact; if above 30, then the re­mainder 30 or 60, being substracted is the Epact for that year. Example for 1661.28 the epact for 1660, and 11 being added makes 39 from which take 30, and there remaineth 9, for the Epact for the year 1661 the thing required.

Note that in orderly counting the years, when you come to the Leap-year, you must neglect or slip one, the [Page 167] reason is, because every Leap-year hath two dominical letters, and there also doth the week day change in the first of March, so that for the day of the month, in finding that the trouble of remembring the Leap-year is avoided.