The uses of a QUADRANT FITTED For daily practise. Both with the ordinary lines for the Hour and Azimuth, and other things of the Suns course in reference to the Horizon. And also with new Lines serving to the fore-mentioned and other purposes more accurately. As namely, To find the hour of the Night by the Stars; To describe the most usual sorts of Dials; To perform all common things in Mensuration. And many other requisite conclusions. Performed In an accurate, easy, and delightful way. By SAM. FOSTER, Professor of astronomy in Gresham college. PUBLISHED by A: T. LONDON: Printed for Francis Eglerfeild, and are to be sold at the Marygold in Pauls-Church-Yard; and by A: Thompson in Hosier-lane, near Smithfield. 1652. AN Advertisement to the READER. COncerning the structure and use of this Quadrant, the Reader may understand thus much. The hour and Azimuth lines are like those that are commonly seen upon other Quadrants, and the uses are( most part) the same, and therefore are lightly passed over; as is seen in the second Proposition. But the Distance of the equinoctial and tropics is here shortened, that so more room might be gained above, for the better placing, and the more accurate dividing of the Equinoctials; which in small Instruments may receive each second degree, in larger each single degree or more. If it be required to make these yet larger, then may the fore-mentioned Azimuth lines be left quiter out. For the use of them, as they are here described, is of small moment, very hardly, making good the Suns cost to one entire degree: and for serious practise, the new Lines added are far more sufficient. If this be granted, then may the Equinoctials stand below, by which means they shall become large enough, even in small Instruments. Especially this may most fairly be done, if the hour-lines be reverted by changing the places of the equinoctial and tropics; that is, if the equinoctial altitudes be inserted below on the circled nearest the limb, and the tropical altitudes above in the circled nearest to the center. Thus becoming more large, they will supply all intended purposes very well. There is no Scheme given of this change now mentioned, nor of the vulgar Hours and Azimuths, because those lines are well enough known already, and this mutation is easy to be understood. If other Quadrants were thought complete in use, this will be found much more copious. For it servetb not onely to find the hour of the day by the Sun: of the night by the stars, and what else belongs to their Risings, Settings, Amplitudes, &c. but is very well fitted also to describe all the most usual sorts of standing dials; that is, all that are upright, or else reclining or inclining to be full East and West: which two sorts will furnish many kinds of such bodies, as are regularly formed. These are here performed by very easy and familiar ways of working. The nocturnal for the hour by the Stars, is more expedient in this then in other Quadrants. For in judging of time only by the Appulse of the Stars to the Meridian, and finding that Meridian too onely by a rude conjecture from the North star, an error of a quarter or half an hour is easily unawares committed. This cannot be so here if any ordinary care be had in taking the stars altitude. For this purpose, there are twelve select Stars inserted, all of them of North Declination, lying between the equinoctial and tropic of Cancer; and in such difference of right ascensions, as that one or other of them will be always in such convenient place of the Heavens, ●● from whence the hour may very fully be collected every night throughout the whole year. Since therefore they are so convenient for use, there would be a little the more diligence used to come to the knowledge of them in the Heavens, that due observations may be made whensoever any of them shall be in view. If any desire that other stars( such as are better known to them) should be inserted, they may have their desire easily fulfilled. Onely they must take care, that the stars be such as fall between the tropics and the Heavens, and chiefly between the Equin●ctial and North tropic; because such stars are longest in view, and their hours best found— The Propositions that are here set down might have been increased both in number and in variety of performance, if prolixity had been affencted; but such of them, and such ways of effecting them, are here pitched upon as seemed most conducible for daily use, and to make the Treatise, rather material ▪ then burdensome. And for the same reason it is, that the several lines upon the Quadrant, are denoted by letters only; that by such brevity, all unnecessary circumlocution might be taken off, which, by imposition of names to each of them, could not so easily have been avoided. If other Quadrants have heretofore found good acceptance, because they were of some good use, I should in reason expect a greater proportion of thanks from the ingenious for making public this larger Improvement of this Instrument But be that as the READER listeth. These additional lines were invented, and the uses written at the request of a Friend, and are intended as an addition or Appendix unto Master Gunters Quadrant, it being most proper for the same, onely some few are published alone in this volume, for the private satisfaction of some Friends, of the Judicious Author Mr. SAMUEL FOSTER. Thine, A: Thompson. depiction of a quadrant The uses of the QVADRANT. I. To find the Suns Declination. LAy the thread to the day of the month upon the back side of the Quadrant, and it will show you the Declination of the Sun in that unequal scale, which is numbered with twice 23½. If your day fall in the upper scale of moneths( which may be called the Summer scale) then is the declination North: if it fall in the lower( or Winter) scale, the declination is South from the equinoctial. Thus upon april 20, you shall find the sun to decline 15 gr. northward: and January 30, it declines about 14½ gr. southward. ¶ The contrary work is easy: by assigning the suns declination, to know on what day of the month the same shall be. For the thread may be laid to the declination in two places; in both which it will cross the two half years, showing two several dayes on which the sun shall have so much declination North; and two more dayes on which it shall have that declination Southward. It will be easy to distinguish which of these days serves your purpose, by the two seasons of the year unto which the two scales of moneths do answer. II. To rectify the Bead for observation of hour or Azimuth: and to perform those things that are done by the usual lines upon the Quadrant. HAving found the Suns declination for your day, you must count the same upon the double equal scale which is on the foreside of the Quadrant, namely, from the middle of it, towards the right hand, if the declination be North, or towards the left hand if it be South. The thread being laid thereto, you must move the bead, till it fall justly upon the hour of 12, so shall it be set right for the intended uses of that day. As, 1 For the hour. If you observe the suns altitude( by letting the sunbeams to shine through the sights, and the Plummet to hang at full liberty, close to the plane of your quadrant) the bead will show the hour, if you have respect to the time of the year. That is; If the suns declination be North, the bead shows the time of the day among the Summer houres, those which spread from the equinoctial towards the right hand. If the Sun decline South, the time must be accounted in the cross lines which are the winter hours. And in this observation you shall see the thread to cut( in the equal limb) the Suns altitude above the Horizon.— Thus at London, if the ☉ decline 15 gr. Northward, and the altitude were 95 / 12 gr. the hour would be about a quarter before 6 in the morning: Or a quarter past 6 in the evening. But if the Sun had the same declination Southward, and the same altitude also, then would the time be half an hour past 8, in the morning; Or half an hour past 3, in the evening. The former of these times is shewed by the bead among the summer-houres; the latter among the winter-hours. 2 For the AZIMUTH. If the suns altitude be numbered the contrary way in the equal limb, and the thread be laid thereto, the bead will then show the Azimuth of the Sun if you account it according to the time of the year: That is, among the Summer Azimuth; when the sun hath North declination; and among the Winter Azimuths when the sun declines South. The Summer Azimuths, are those that spread from the equinoctial towards the left hand: The other crossing them, are the Winter Azimuths.— Thus if the suns declination were 8 gr. Northward, and the altitude 18 gr. the Azimuth would be 80 gr. from the South. But if the sun had 8 gr. of South declination, and 18 gr. altitude, the Azimuth would be 50 gr. from the South, here at London. This way may serve for gross works, when the Azimuth is required onely within one or two whole degrees. You shall find it done more accurately and for better purposes in the thirteenth following. 3 For the ascensional difference. The bead being( rectified as before) and applied to the left side of the quadrant, gives the ascensional difference, or the time of sun-rising & setting before or after 6 a clock, among those hours and quarters which intersect each other upon the same left side of the quadrant, if you account them agreeable to the time of the year. And from the bead to the line of 12 rightly taken according to your time of Summer or Winter, gives the semidiurnall Ark of the sun, or half the dayes length.— As also, from the bead to the other line of 12, which serves for the contrary time of the year, gives the seminocturall Ark, or half the length of the night.— Thus if the suns declination were 14⅓ gr. the ascensional difference would be 1 hour and ¼ of an hour. And if the said declination were North, then the sun riseth that day ¼ of an hour before 5; setteth ¼ after 7: The semidiurnall ark( from the bead to the Summer 12) is 7¼ hours. The seminocturall ark( from the bead to the Winter 12 is 4¾ houres. These doubled, make the day 14½ hours long, The night 9½ long. 4 For the AMPLITUDE. The bead applied to the right side of the quadrant gives the Amplitude of sun rising and setting, in all varieties. Namely; From the bead to that South Azimuth which is proper to the season of the year, is the Amplitude from south: as also to the contrary south Azimuth, gives the Amplitude from North: showing how many degrees of the Horizon the sun riseth and setteth any day from the just South or North. So from the bead to the East and West Azimuth ( which is the ninth Azimuth) gives the Amplitude from East or West.— Thus if the ☉ decline 14⅓ gr. The Amplitude is here, 23½ gr. almost. If the declination be North, then is this Amplitude from East and West toward the North, 23½ degrees. The Amplitude from the north itself is then 66½ gr. From the south point of the Horizon it is 113½ gr. You may easily( in such manner) account it for South declinations of the ☉. V. To find when Twilight begins in the Morning and ends at Evening: which moments are the two utmost terms of dark night. AFter the bead is rectified for your day, The thread laid to 18 gr. in the equal limb, will show the hour or part required. Onely here remember to take your hour aright. Namely, in winter time, look among the Summer houres, where it is that the bead resteth, for that is the Morning or Evening hour of Twilight. So in summer time you must look among the winter hours.— Thus when the ☉ declines 111 / 10 gr. Southward, the twilight begins at London at 5 in the morning, and ends at 7 a clock at night, as the bead shows among the summer houres. But if that declination were North, the twilight would begin at ¼ of an hour before 3 in the morning, and end ¼ after 9 at night.— The suns depression 18 gr. under the Horizon, is the usual term whereon to begin and end the twilight. You may as well do this to any degree of light, as to 12, or 13 degrees depression: At which time in the morning all things begin to be visible, and the light to be of some use. As if the ☉ decline 3½ gr. Southward, if you set the bead thereto, and then lay the thread at 12 gr. in the equal limb, you shall see the bead( among the summer houres,) fall upon 5, in the morning and 7, at right, So that at 5, and till 7, there is a reasonable degree of light. Or if in summer the ☉ had declined 7½ gr. Northward; the said degree of light would begin at 4, in the morning, and end at 8 in the evening.— near to the longest days you shall find no twilight at all according to 18 degrees depression of ☉ under the Horizon: for then the bead will fall be the Winter 12 a clock-line. ¶ These are the chief uses of the hour and Azimuth lines as they are here, and in all Quadrants commonly inserted. There are other things concerning the suns place in the ecliptic: The suns declination: The suns right ascension: Namely,— How by having any one of these, to find out the rest— These are here omitted as matters onely of curiosity, being of no further use, in this instrument, then that they may be known. Yet if any should desire them, they may have a Scale of the 12; signs inscribed on the back side; by help of which, the fore-named requisites may be attained. The particulars that follow are most aimed at,( as being more of them, and more accurate) and therefore the precedent things are thus briefly passed over. III. To find the Suns ascensional difference, &c. COunt the declination in the equal limb from F. to K. The thread there laid, gives BS the ascensional diffeence.— The said ascensional difference gives the times of sun rising and fetting before and after 6: with the lengths of Day and Night.— The same may be done for all stars whose declinations are known. ¶ So by having the ascensional difference, you may find the suns declination thereto belonging. Here at London, if the declination be 20 gr. the ascensional difference is 27 gr. 14 min. That is, 1 hour, 49 minutes. And if this declination be North, the sun riseth 1 hour 49 min. before 6, and setteth so much after 6. That is, it riseth 11 min. after 4 in the morning; and setteth 49 min. after 7 a clock at night. And the time of setting being doubled, gives 15 hours, 38 min. for the dayes length. The time of rising being doubled, gives 8 hours, 22 min. for the length of night. But if the declination had been South, the sun should rise 1 hour, 49 min. after 6( that is, at 7, 49 min.) And should set 1 hour, 49 min. before 6( that is, at 4 and 11 min.) and the day would be 8 hours, 22 min. long; the night 15 hours, 38 min. IV. To find the suns Amplitude, &c. COunt the declination in the equal limb from G to H. The thread there laid, gives C R for the Amplitude.— The same may be done for stars whose declinations are known. ¶ So by having the Amplitude, you may find the declination. For if the Amplitude be counted from C to R, the thread laid at R, gives the declination G H. At London, if the declination be 20 gr. the Amplitude is 33 gr. 20 min. from the East and West points of the Horizon. V. Having the declination of any upright plane, to find the elevation of the style, &c. LAy the thread to the planes declination, counted from D to R: So will G H be the Elevation. ¶ So by having the Elevation G H, you may find D R the declination. If an upright plane( here) decline 20 gr. the styles elevation will be 35 gr. 48 minutes VI. To find the Deflexion, &c. COunt the declination from B to S. The thread there laid, gives F K the Deflexion. ¶ So by having F K the Deflexion, you may find B S, the planes declination. If A plane declining 20 gr. the Deflexion is 15 gr. 13 min. VII. To find the planes Difference of longitude, &c. 1 COunt the Elevation from F to K. E S is the difference of longitude. 2 Count the Deflexion from G to H. C R is the difference of longitude. ¶ By the contrary works, having the difference of longitude, you may find the Elevation and Deflexion. A plane declining 20 gr. hath 25 gr. diff. of longitude. VIII. To make an horizontal dial. 1. COunt the hour from E to S: The thread laid at S, gives F K. Then count G H equal to F K: The thread there laid gives D R, the space of that hour from 12. 2. Count the hour from C to R, and by help of the thread you shall have G H. Then count FK equal to GH: the thread laid at R, gives B S, for the space of that hour from 12. 3 With a pair of compasses, take the hour from C to R, and set it from B to S. B S is the space or angle of that hour from twelve. 4 Take with your compasses the hour from E to S, and set it from D to R. So the number D R shows how many degrees that hour must be from 12. By all these ways( here at London) the third hour will be found about 38 gr. from 12. The rest will be in like manner found according to their true quantities. IX. To find what Angle any hour-circle maketh with the Horizon; or any Azimuth makes with the equinoctial. LEt the number of the hour-circle( or Azimuth) from south, be counted from C to R the thread laid at R, will cut the equal limb in H. And F H will be the angle required. ¶ By the angle know, it will be easy by the contrary work, to find the hour( or Azimuth) to which that angle belongeth. The third hour( or 45 Azimuth) makes with the Horizon( or with the equinoctial) an angle of 63 gr. 53 minutes here at London. X. To find what ark of any houre-circle is intercepted between the Equinoctiaell( or any Parallel) and the Horizon. COunt the number of the houre-circle from South, from E to S: or if it be above 90, from E to B, and back again to S. So F K in the equal limb, will be the ark required, between the equinoctial and Horizon. The ark intercepted between any parallel and the Horizon, may hence also be found.— If the Declination of the parallel be North, and the hour be between 12 and 6, Add the declination to the ark found by the former work: In other houres beyond 6, subtract the former ark out of the declination, the result will be the ark required. Upon the hour of 6 itself, the declination of the parallel is the ark intercepted.— If the declination be South, Subtract it out of the ark found before,( namely, the ark intercepted between the equinoctial and Horizon) what remains is the ark intercepted between that parallel and the Horizon. Thus at London. The ark of the 3 hour intercepted between the equinoctial and Horizon, is 29 gr. 21 min.— And if the declination be 18 gr. North, the ark intercepted between that parallel and the horizon, is 47 gr. 21 min.— If the parallel be 18 gr. South, the ark will be 11 gr. 21 min. ¶ The first work, will also show what ark of any Azimuth from South is intercepted between the Horizon and equinoctial, if in stead of the hour-circle from South, you use the Azimuth from South. This intercepted ark, is the equinoctial altitude of that Azimuth. So in the 45 Azimuth from South, the equinoctial is 29 gr. 21 min. high. In the 135 Azimuth from South, the Equinoctialls depression under the horizon, in 29 gr. 21 min. This is made use of afterwards. XI. How high the Sun shall be upon any Azimuth, and in any Declination. THe Azimuth is best numbered from the south. And this proposition( with most of those that follow) is to be done by help of compasses. ¶ If the ☉ be in the equinoctial, the first work of the last proposition gets the equinoctial altitude or depression, by counting the Azimuth from E to S, whereby the ark F K will be found. This ark( if the Azimuth be less than 90) is the altitude: if more than 90, it is the depression. But if the sun have Declination, then First, lay the thread from F towards K according to that Declination, and take the least distance from the point B, to your thread, and keep this extent. Then, ¶ If the suns declination be south; Count your Azimuth from E to S, and lay the thread there, which will cut the line E N, in T. Set one foot of the former Extent in T, and turn the other about toward the side A B, applying the thread to the remotest distance of that circuit. The thread so laid, will give the altitude required; if you count the degrees from F. Thus the sun declining south 11 gr. 30′ will have 16½ gr. of altitude in the 45 Azimuth. ¶ If the suns declination be North,— and the Azimuth less then 90 from south; Count your Azimuth from E to S, and lay the thread at it, and let it cut E N, in T. Then set one foot of your former extent in T, and with the other foot turned about, lay the thread at the remotest distance, from T towards the side A C. The theed so lying, shows from F in the equal limb, the altitude required. Thus if the sun decline 11½ gr. North, his altitude upon the 45 Azimuth will be 42⅕ gr.— But if the Azimuth be more than 90, count from B to S, the excess above 90; and applying the thread thereto, see what degrees of the equal limb the thread cuts from From F. Count that number of degrees from 60( in the equal limb) forward, toward 70, 80, 90, and lay the thread there, which suppose to cut the line α ω, in π. Set your compass( keeping still their first extent) upon π, and turn the other foot towards the the side A C, laying the treed at the remotest turn. If now to the thread so laid, you number the degrees in the equal limb, from 60, the same shall be the altitude required. Thus if the sun decline 11½ gr. North, and the Azimuth be 101¼ gr. from South, the altitude must be 5¾ gr. in our latitude of 51 gr. 30 minutes. Another way for this Proposition. BY the first work in this 11th, get the Equinoctial altitude or depression for your Azimuth. Then lay the thread at E: and in C D, from D, count the said altitude or depression; from which number or point, take the least distance to the side A C. Enter this length between the side A C and the thread, keeping one foot upon the line A C, remove it thereon to and fro, till the other foot turned about may justly touch the thread. Then keeping your compasses there set, remove the thread from Gtoward H, according to the Suns declination, and take the least distance from your former standing to the thread. This length measured in the Scale CD[ so as one foot standing upon the Scale, the other turned about may justly touch the side A C] shows an ark, Which, If the suns Declination be south, must be substacted from the Azimuths equinoctial Altitude. If the suns Declination be north, and the Azimuth less than 90, must be added to If the suns Declination be north, and the Azimuth more than 90, the Azimuths equinoctial depression must be taken out of this ark; The result is the altitude looked for. Thus if the Azimuth be 70 / 110 from south, the equinoctial Altitude / Depression will be 15¼ gr. The ark found, will be 14¼. Then, If the ☉ decline 11½ south, the altitude upon the 70 Azimuth will be 1 degree. If the ☉ decline 11½ north, the Altitude upon the 70 Azimuth will be 29½ degrees. If the suns declination were 20 gr. north, That forementioned ark would be 25 gr. whence taking 15¼, there remaines 9¾ for the Altitude of the sun upon the 110 Azimuth from south, at that Declination of 20 gr. north. ¶. By this work, may a table of altitudes be made, by which the former Azimuth lines upon the quadrant may be inserted. XII. To find how high the Sun shall be at any hour, and in any Declination. FIrst, find the intercepted ark of your hour, between the parallel of Declination, and the Horizon, by the tenth. Secondly, find what angle your hour-circle maketh with the Horizon, by the ninth. Thirdly, count that angle from C towards D, and from thence take the least distance to the side A C. Measure this length upon the side A C( from A) and there set your compasses. Then keeping that station of your compasses, Lay the thread to the intercepted ark, counted in the equal limb from G: and take the least distance from your standing to the thread. Set one foot of this length in the scale C D, so, as that the other being turned about, may touch the side A C, so shall that foot in the scale C D, give the Degrees of Altitude required, if you number them from C. Let the hour be three from Ncon. The intercepted ark between the equinoctial and Horizon, will be 29 gr. 22 minutes. And if the Sun decline North / South 11½ gr. The intercepted arkes will be 40.52 / 17.52. And the angle of the third hour with the Horizon, is 63 gr. 53 ▪ So that the Altitude for North / South Declination of 11½ gr. will be 36 / 16 degrees. ¶ By this work you may make a Table of the Suns altitudes upon any parallel of Declination. And by those altitudes you may insert those Summer and Winter houres which are upon the Quadrant. XIII. To find the Suns Azimuth. FIrst, lay the thread to the Suns Declination counted in the equal limb from F to K, and take the least distance from the point B, to the thread, and keep your compasses at that extent. Then count the Suns Altitude in the equal limb, from F, and lay the thread to it. This being done, ¶ If the Sun Decline South, keep one foot of your compasses always upon the line E N, beyond the thread, towards E, and remove it still upon that line, till the other foot being turned about may touch the thread precisely. Observe then, where the foot of your compass standeth upon the line EN suppose at V. Bring the thread to V, and it shows( from E) the Azimuth from the South. ¶ If the Sun Decline North, keep one foot of your former extent, upon the line E N, on this side the thread, towards N, and remove it still upon that line, until the foot that is turned about do touch upon the thread. And observe where your compass foot then standeth, upon the line E N( suppose it stand at W.) Lay the thread at W, and it will cut the scale E B. The parts whereof, from E to the thread, are the Azimuth from South. But if it so fall out in North Declinations, that when the thread is laid to the altitude, you cannot find room upon the line E N, whereon to set your compasses so as to keep the conditions before required; then work in this manner. Add always 30 degrees to the Suns altitude, and lay the thread at that compound Altitude, numbered in the equal limb from F. To the thread so laid, enter the former extent of your compasses, between the thread and the line α ω, keeping one foot always upon that line. And look where the foot of your compasses resteth upon that line; suppose at π. Take then the length from π to α, and set it upon the line N E( from N towards E): and to the point wheat it rests, apply the thread: observing what parts it cuts upon the scale, from B. The number of those parts, gives the quantity of the Azimuth above 90 from the South. Or the parts cut from E, give the Azimuth from the North. ¶ If the ☉ decline not at all, but is in the equinoctial, then the sole Altitude from F to K( by help of the thread thereto applied) gives E S the Azimuth from South. If the Altitude of the ☉ be 21⅔ in the equinoctial, the Azimuth from South is 60 degrees. If the Sun decline South 5 gr. and the Altitude were 15¾ gr. the Azimuth would be 60 degrees. If the ☉ decline North ●0 gr. and the Altitude were 50, the Azimuth would be found 50 gr. If the ☉ decline North 20 gr. and the Altitude were 9¾ gr. the Azimuth would be 110 gr. from the South. ¶ If you suppose the sun to have no Altitude, and do work by these rules, you shall find the suns Amplitude, Ortive and Occasive, from the South. As if the sun Decline 20 gr. North, you will find 123 gr. 20 min. for the Amplitude from the South. XIIII. To find the hour of the day, by the Sun. COunt the suns Altitude in the equal limb from F: and to the thread there laid, take the least distance from the point B: and keep this distance. Then count the suns Declination( which is had easily by the first proposition:) from F in the equal limb, and apply the thread, to it. Then further, ¶ If the declination be South, set one foot of your former extent, upon the line E N( always on that side the thread on which E standeth from it) and remove it thereon, till the other( turned about) may justly touch the thread A K. Suppose( in so doing) the compass foot stayeth at V. The thread applied to the point V, will cut the hour from Noon, if you count the intercepted parts upon E B, from E.— Thus if the sun decline 20 degrees South, and the Altitude were 13 gr. 50 min. the hour at London would be 10, or 2. ¶ If the Declination be North, set one foot of your former extent upon the side A C, removing it thereon to and fro, till the other foot turned about, will onely touch the thread. When it is so fitted, let that foot upon the side A C, keep its station; and from thence extend the other foot to the suns Declination counted in the scale A P This last extent must be applied to the line N E, from N: and where it stays, lay the thread. So the parts cut upon the scale E B, will give the hour.— But this must be done with caution For if that foot that kept its station, stood from A, beyond the Suns Declination in the scale A P, then the intercepted ark from E to the thread, gives the hour from Noon. But if the forenamed foot stood between A and the Declination, then the whole ark E B 90, with the ark from B back again to the thread( these two put together) give the hour from Noon. Thus, if the sun decline 15 gr. Northward, and be 21 gr. high, the hour is 7 before, or 5 after, Noon. Or if the altitude were 2⅔ gr. the hour must have been 5 in the Morning, or 7 in the evening: namely, 90 and 15 degrees from Noon. XV. On an Upright declining plane, to find the angle between 12 and 6. COut the palnes Declination from C towards D: From that point take the least distance to the side CA. Set that length from M to Y, upon the line M Y. The thread laid at Y gives G K for the angle between 12 and 6. Or count the Declination of the plane from B towards E, and lay the thread at it. The thread will cut N E. Take from N to the intersection, and apply it to M Y; the thread put to Y gives G. K, as before. If a plane Decline 20 gr. this angle will be 66¾ at London. XVI. To find the Declination of a plane. FIrst, draw an horizontal line upon your plane( which you may do by your quadrant.) Then apply one side of the quadrant to that line, so as the limb may be toward the sun, and the plane of the quadrant may lie Horizontally flat. Thirdly, having a loose thread and plummet, you must hold that thread close by the edge of the limb( letting the plummet hang down at liberty) till the shadow of the thread passeth directly through the quadrants center. Which done, you shall see what degrees of the limb the shadow cuts from that side of the quadrant which is perpendicular to the horizontal line. This is called the horizontal distance. At the same moment of time, observe the same Altitude. By this Altitude you may get the suns Azimuth from South, by the thirteenth. After this preparation, take diligent notice, whether the shadow of the thread fall betwixt the South and the perpendicular side of the quadrant, Or whether the same shadow fall so, as to leave both the South and the said perpendicular side( both of them) upon one cost of the shadow. In the first case, you must add the horizontal distance to the Azimuth. In the latter case, you must subtract the lesser out of the greater. The result( whether it be sum or difference) gives the planes Declination from the South. Note here in the second case. That if the horizontal distance be greater then the Azimuth, then doth the plane decline to that cost( East or West) which is contrary to the cost on which the sun stood from the South. This falleth our very frequently. Note also in the first case. That if the sum of the horizontal distance and Azimuth do exceed 180 gr. then the planes Declination from South, is contrary to that cost whreon the sun stood. And it is found, by substracting the forementioned sum out of 360 degrees. This happens more seldom: that is, onely upon some North planes; and on them, onely then, when the suns Azimuth is more than 90 from the South; and the horizontal distance more than is the Azimuth from the North. Examples are here omitted for brevities sake. Onely add this. That if the planes Declination from South be above 90 gr. you must subduct it out of 180, and the remainder is the Declination from the North.— By this accounting from North and South, you may always make that your plane decline not above 90. And as when it declines nothing it is a full South or North plane; So if it decline just 90, it is then a full East or West plane. XVII. How to draw any upright declining dial. FIrst, draw a perpendicular or Plumb-line A B, and cross it at right angles with the horizontal line B C: and make B A equal to A {us} in your Quadrant. 2 Upon the equal limb of your Quadrant, count the planes declination( from North or South) from G, and there keep the thread: which will cut some of those lines that are drawn within the upper square. 3 Observe first, those intersections which the pricked lines make with the thread at b, d, m.— Take then the length from A the center of the quadrant to b; and set it here upon the horizontal line from B to 1,( always on that side of B, which looks to the same cost whereunto the plane declineth.) So, take from the quadrants center A, to the second pricked lines intersection with the thread, at d; and set it here from B to 2. So likewise the third A m; must be set from B to 3. 4 Observe again all such intersections as are made with the thread, by the rest of those lines whose common concurrence is in the point M, namely, at a, c, e, h: and take their several lengths from the quadrants center A, and prick them here down on the other side B( contrary to the cost of declination) namely, at 11, 10, 9, 8. Then for the next line upon the quadrant( which doth not, but would intersect the thread, if it were drawn out far enough) observe where the thread cuts the extravagant line r s, namely, in s: and take from A to s, and turn that length twice from B, so shall it design the point 7. Afterwards at the point 7, draw the infinite line CD parallel to B A. Also set off the hour of 6, on that side B which is contrary to the cost of Declination, namely, from B to E, according as the angle between 12 and 6 shall be found by the fifteenth. 5 Draw all the hour-lines from A, the Declination 28d. S. East diagram center of your dial, through the points 3, 2, 1, 12, 11, 10, 9, 8, 7, in such wise, that as many as well can, may cut the line D C, as is here done, in p and q. 6 Make 6, 5, equal to 6, 7: and 6, 4, equal to 6, p: and 6, 3, equal to 6, q: and draw the rest of the houres, A 5, A 4, A 3. Thus you may get 12 houres, and if you extend them beyond the center, you shall have the whole 24. Out of which you may make choice of such as will serve your use. ¶ For placing the style, Seek the Elevation and Deflexion by the fifth and sixth. And make B F equal to the Deflexion, setting the substylar line F A always on that side 12 which is contrary to the cost of the planes declination. Make also F G equal to the Elevation: So F A G will be the pattern of the style. Or the thread lying still at the planes declination upon the Quadrant as it did, Take the least distance from the point X to the thread; and set that length from B to H, and draw A H for the Substylar. Then making A H K a right angle, take the least distance from M to the thread, and make H K equal to this distance: So is K A H the pattern of your style. ¶ In all Dials, The style must stand just over the Substylar, Elevated so much above it, as the Elevation( before found) cometh to. In South Upright decliners, the center of the dial is above( as in the former figure) and the style points downward. But in North decliners, the center must be below, and the style must point upward. XVIII. Of the upright full South-Diall. THe Declination of the full South dial is nothing. Whence it is, that The angle between 12 and 6 is 90 degrees. The line of 12 is the substylar. The styles Elevation is the compliment of your latitude. The way of pricking down the houres is( in a manner) the same with that before for decliners. No more needs ●● be said of it. The Erect full North plane is the same with this South. Onely the style of this, points upwards toward the North pole, as the former doth downward towards the South pole. XIX. Of Upright far declining plains. THese dials are more difficult than those other decliners mentioned in the seventeenth, because here the houres have no center or point of meeting upon the plain. It will not be amiss therefore to set down the whole work in all parts of it. 1 Draw a perpendicular or plumb-line A B, and cross it at right angles with the horizontal line B C. And make B A equal to A ☉ in your quadrant, setting A above B if the plain decline from the South; or below B if it decline from North. 2. Count the plains Declination from South or North, upon the limb of your quadrant, from G: and there keep the thread. 3 Among those lines on the Quadrant( whose common concurrence is at M) observe that intersection which is made by the 6th hour from the quadrants center, with the thread. Take the length from the same center to that intersection, and prick it down here from B to C( and on that side B which looketh toward the South, if the plain decline from South: or toward the North, if the plain decline from North.) And draw out the lines C D E parallel to B A. 4 Observe again upon the quadrant that intersection which the second line from the the center makes with the thread, and take the length from the center of the quadrant thereunto, and prick it down towards C, namely from B to F. 5 Take the lengths from the center of your quadrant to every hour point upon the side A C: and prick them all down here, from C to 7 and 5, from C to 8 and 4, from C to 9, 10. And lastly, take from the center of your quadrant to the point r, and turn that length twice from C: this double length will reach from C to 11, at E. 6 Lay a ruler to A and F, and transfer the the point F unto H in the line C E. Then take the length from H to 10, and set it from A( towards B) to 10, the same way from A that 10 stands from H. 7 With the same length H 10, or A 10, go to your quadrant, and setting one foot of it, on the side A C, in the fourth point from the Center, with the other( turned about) lay the thread at the remotest distance, and keep it there. An upright Plain declining 82 deg. from South, Eastward. diagram 8 From every point on the side A C of your quadrant, take the least distances to the thread so laid; setting them down from A to 7 and 5, from A to 8 and 4, from A to 9. A 10 was put on before. Then the least distance from r to the thread being twice turned from A towards B, will give the length from A to 11. 9 For the finishing then of the houres, you have no more to do, but draw right lines through each couple of correspondent points; namely, from 4 to 4; 5 to 5; from C to A, or 6 to 6; from 7 to 7; 8 to 8; 9 to 9; 10 to 10; and from 11 to 11. ¶ Concerning the forming and placing of the style. 10. BY the precedent seventh proposition you may find the plains difference of Longitude, which( for this plain that declines 82 gr.) will be( here at London) 83 gr. 43 min. and that from the South, because the plain declines from the South. The compliment of which longitude( 83 gr. 43 min.) is 6 gr. 17 min. Take then first, the length from C to 7 the next hour point upon C E, and carrying that extent to your quadrant, set one foot of it upon 15 in the scale A P: and lay the thread so, that the other foot turned about may just touch or pass over it: and keep the thread there. Then( in the scale A P) count the forementioned compliment, 6 gr. 17 min. and taking the least distance from that point to the thread, set it from 6 a clock at C, towards E if the plain decline from South,( or towards D if the plane decline from North) as you see it done here, at G. Secondly, do the same work again upon the line A B. That is; Take from A to 7 the nearest hour point, and set one foot of that extent upon 15 in the scale A P, and with the other foot turned about, lay the thread as before. Then in the same scale A P, count the same number 6 gr. 17 min. and taking the least distance from thence to the thread, set that length from A to K, answering to C G. And last of all, draw the right line G K. This shall be the line of deflexion, over which the style must stand. 11 Furthermore. Through the points G and K( or any other two points of the same line) draw the two lines G O, K P, both perpendicular to the deflexion line G K. Then considering that every hour comprehends 15 degrees of longitude( that is, that from C to 7 is 15, and from 7 to 8 is 15, &c.) and since that C G is 6 gr. 17 min. If ● G, be taken out of C 7 which is 15 gr. there will remain G 7, 8 gr. 43 min. To which, if you add from 7 to 9, which is two houres or 30 degrees, the sum will be 38 gr. 43 min. whose compliment is 51 gr. 17 min. If now you make the angles G M R, and K N S, each 51 gr. 17 min. they will cut the Deflexion line G K, in R and S. And if further, to the radius G R, you describe the ark R T; and to the radius K S you describe the ark S V; and draw the line T V, a tangent to both these arkes, the Trapezium G T K V shall be the pattern of your style. In placing which, you must be careful that these perpendicular lengths G T and K V( perpendicular I say to T V the fiducial edge) be justly placed upon the two assumed points at G and K.— Or having found G 7 to be 8 gr. 43 min. you may add to it from 7 to 10, which is( three houres or) 45 degrees. The sum will be 53 gr. 43 min. whose compliment is 36 gr. 17 minutes. If now from the points O and P( where the said hour of 10 cuts the two f●re-mentioned perpendiculars G O and K P) you make the angles G O R and K P S, each equal to 36 gr. 17 min. they will cut the deflexion line G K in the same two points R and S. After which, you may proceed to make the pattern of your style, as before. ¶ 1 Note that in performing the fifth section of this proposition, instead of taking those hour points from the Center of your quadrant upon A C the side of your quadrant( if those distances should be too great for your plain) you may lay the thread any where upon the Quadrant, and instead of taking from the center to the fore-named points, you may take the least distances from the said points to the thread, severally, and set them down from C to 7 and 5, and from C to 8 and 4, and so to 9, 10; and for 11, you must take from the point r to the thread, and set it twice from C; by which means they will all be of less distance from C. And then all the work is to be continued, as is before prescribed.— Or if the said distances should be too little, you may double, triple, or, &c, to make them greater. ¶ 2 Note again, that in decliners from the North, that difference of Longitude which you find by the seventh, is to be reckoned from the North, and so the compliment of it is to be accounted from C( or 6 a clock) towards D. And that the widest parts of the houres in these North plains must point upwards, and the closest parts downward; contrary to what is expressed here in this plane, which hath its Declination from the South. ¶ 3 Note lastly, that this direction here given for enlarging the houres in far Decliners, may easily be applied to such direct, or horizontal Dials( as are mentioned in the 26 follwoing) upon which the pole hath but small Elevation. For the dial( or onely some chief houres of it) being described in its natural straightness, may be enlarged by the same means that this last was. Which will not be hard to do, but would be tedious here to run over again. XX. Of full East and West upright dials. THese are more easy than the former sort were. For having drawn the plumb-line A B, and assumed the point A for the hour of 6; go to your Quadrant, and take from the center of it to all the houre-points upon the side A C; and prick the first of them down in the line A B, from A to 5 and 7: the second from A to 4 and 8; the third from A to 3: the 4th. from A to 2. And for the fifth, Take from the center of your Quadrant to the point, r, and set that length twice from A, so it shall limit out the point 1.— Having these points, draw lines through them, all parallel one to the other, and all pointing up to the North; namely so, as to make the acute angles B A C equal to the compliment of your latitude. A full West upright dial. diagram ¶ For the Style. IT must always stand over the line of 6 o'clock, parallel to it, and distant every where from it according to the length of A D. Which length is soon found, by drawing A D perpendicular to the hour-lines, cutting the third hour from 6, in D. By which line you may make the pattern of your style. For the fiducial edge lies parallel to the line of 6, A C, and at the distance of that line A D. ¶ 1 Note here too; that if your lengths from the Quadrants center to the houre-points be too long, you may shorten them by laying the thread upon the Quadrant according as your convenience shall direct, and taking the least distances from those houre-points to the thread; and so pricking them on from A or 6, to 5, 4, 3, &c; as was before mentioned in the first Note upon the former proposition.— Or if they be too little, they may be doubled, &c. as is there expressed. ¶ 2 Note further, that what is here done for describing these East and West dials, may be applied to the direct Polar plain. Onely remember that you are not tied( in the Polar) to make the houres to any set angle with the line B A, but they are best at right angles; for then the line A B may be taken for, and placed as, the horizontal line of the said plain; all the houres lying as vertical lines unto it. And also the line of 6 here, must be taken( in the direct polar) for the line of 12, and the rest of the houres are to be drawn alike on both sides 12: nothing in substance differing from these East and West Planes. XXI. In East and West re-in-cliners, To get the Deflexion. COunt the re-in-clination from D towards C. Take the least distance from thence to the side A C. Set that length ●rom M to Y, and lay the thread at Y. The ●egrees F K will give the Deflexion. The substylar line must ascend in Recliners, ●nd descend in Incliners, from the line of 12, according to the quantity of this Deflex●on. The line of 12 lies always Parallel to the Horizon. XXII. To find the angle Between 12 and 6. COunt the Re-in-clination From E towards B. The thread there laid will cut the equal limb. The degrees whereof from G to the thread, are the angle required. XXIII. To get the Styles Elevation. LAy the thread to the Re-in-clination numbered in the equal limb from F, and take the least distance from N to the thread. Set one foot of that length in B, and ●ay the thread so as to touch the other foot when it is turned about. The thread so laid, gives the Elevation in the equal limb, ●rom F. XXIV. To find the difference of Longitude. 1 COunt the Deflexion in the equal limb from F, and lay the thread to it; and take the least distance from B to the thread. Put one foot of this length in N, and apply the thread to the remotest distance of the other foot. The thread will then show in the equal limb, the difference of longitude, if you count from F. 2 Count the deflexion in the equal limb, from G: and to the thread there laid, take the least distance from B. Measure that length upon the side A B, from A; keeping one foot there fixed. Then lay the thread to the plains Re-in-clination counted also from F in the equal limb, and take the least distance from your standing to the thread. Set one foot of this length in B, applying the thread to the other foot turned about. The thread so laid, gives the difference of Longitude in the equal limb, from G. Thus if an East or West plain Re-incline, here at London, 30 degrees, it will have in Deflexion 47°. 26′. Angle from 12 to 6. 55. 26. Elevation 23°. 02′. Differ. of long. 70. 14. XXV. How to draw the dial. VPon the backside of your Quadrant, in the upper part of it, you have lines drawn altogether like those on the foresaid placed near the Quadrants center, the use of which was shewed before. The manner of work in this proposition is in most things suitable to that in the seventeenth, and will need no other direction. Onely for placing the lines, Take notice, that, The line of 12 in these East and West Rein-cliners, lieth always parallel to the horizontal line of the plain. So that if we suppose the former figure of the seventeenth to represent one of these dials, then A B must be conceived to lye horizontal, and B C vertical. All other works will be like to those in the seventeenth. The style in recliners pointeth upward, and the substylar and hour of 6 do ascend above the line of 12, So much as the Deflexion and angle from 12 to 6, come to. The center of the dial is on the South end of the line of 12. The style in incliners pointeth downward, and the substylar and hour of 6 do descend below the line of 12, so much as the Deflexion and angle from 12 to 6 come unto. The center of the dial is on the North end of the 12 a clock line. These things being observed, you must count the Re-in-clination of your plain in the equal limb on the backside from the left hand toward the right, according as the figures are set: and there lay the thread and keep it. Then observe how it cuts the lines next to the center, and proceed in all things as in the seventeenth before. ¶ Note that you may find the inclination of a plain by applying one side of your Quadrant to the plains vertical line: for so the thread will cut the quantity of inclination in the degrees of the equal limb being numbered from that side of the Quadrant which toucheth the plain.— And for finding the reclination, you may lay a ruler to the vertical line of the reclining face, and take the inclination of the under side of that ruler. That inclination will be the same with the reclination. Note also, that this here delivered for East and West Re-in-cliners, is intended chiefly for drawing houres upon those kindes of plains when you meet with them upon Bodies cut regularly. For otherwise you will hardly ever find any such just plain upon a fixed building. Lastly, for a Scale of chords, which here, and in some of the precedent precepts is required, you may make use of the equal limb of your quadrant. XXVI. To make an horizontal dial to any Latitude. FIrst, draw the right line B C, and erect the Perpendicular A H. Then take from the center( on either side of your quadrant) to the third hour upon the side A C; and make A H equal thereto. And draw F H parallel to B C; and the line 5 K 7 also just in the midst of them.— After this, lay the thread to the Latitude of the plain counted in the equal limb: and take from every point of th● side A C, the least distance to the diagram thread, and set each of them down both ways, namely, from A to 4 and 8, from A to 3 and 9, to 2 and 10, and from A to 1 and 11. Then take from the point r upon the side A C, to the thread, and set that length from K to 5 and 7, both ways.— You have now nothing more to do, but onely from H to draw the hour-lines to all the forenamed points: so the draft is easily finished. The style must stand upon the line of 12, and is to be elevated according to the plains latitude: as the manner is in all horizontal dials. ¶ The use of this proposition is to draw all dials in any Latitude for any direct re-in-clining plain. For, the re-in-clination compared( in North re-in-cliners) with the poles Elevation: or( in South direct re-in-cliners) with the Equinoctialls Altitude, will easily give the plains Latitude: in the former, the difference was the Elevation itself: in the latter, the compliment of the poles Elevation.— And this proposition, with the seventeenth for upright plains; the twentieth for upright East and West, and so also for Polar plains on which the pole hath no Elevation: the twenty fifth for East and West re-in-cliners: the eighteen for full North and South erect; will furnish you with ways to draw dials upon such regular bodies, whose plains have any such of the fore-mentioned Aspects. XXVII. To find the hour of the Night by the stars. THe Stars upon the Quadrant( one or other of them) will always be in convenient place of the heavens: that is, of two or more houres distance from the Meridian.— Having then made choice of that star that is fittest, look what number is annexed to the name of it. seek that number in the left margin of the foreside of your quadrant, and close by the hour lines, and rectify the Bead to it.— Then hold up the quadrant steadily, with the sights leveled to the star, as if you were to take the stars Altitude: and you shall find the Bead to show( among the Summer houres of the quadrant) the motion of the star in houres, quarters, and parts of a quarter. This is called the stars hour; but this is not the hour of the Night till it be turned into the Suns hour: which thing is to be done in this manner. Look upon the back-side of the quadrant for your star, and lay the thread upon it: slipping the Bead down to the slope houres below, till it stand upon the same quarter and part( from some just hour on the left hand of the bead) with the stars hour before found. Then note the said hour on the left hand which goeth next before the bead, for that must be supposed to represent the stars hour, and must therefore be called by the same, or number that the stars hour was. And following houres( from the Bead towards the right hand, must successively take their numbers, until you come to be under the day of your month. Unto which day if the thread be laid, the bead will( by keeping your former account) show the true hour, quarter, and part, of the Night. Example, 1. On January the 20, the hour of Cor Leonis was observed Eastward of the Meridian, to be 9 and ¼ and l⅓ part a quarter. The thread laid upon that Star, on the back-side of the Quadrant, will cross the slope houres as doth the line A B. And the bead put down to the forementioned parts of the hour, will stand at the point B. So that the hour C must be called 9 a clock, which is the observed hour of the star. Then the line D must be called 10 a clock: and the thread being put to January 20( taken in the lower circular line of moneths) will lye in the line A E; and the Bead at E shows the time of the Night to be past( the line D, that is, past) 10 a clock, about ¼ and l⅓ part of a quarter, which is 15 and 5 minutes: or 20 minutes past 10 at night.— But if this observation had been upon the second day of November: then the thread laid upon( the day given in the lower circled of moneths) November 2, would lye in the line A F: and the Bead would be upon the full houreline that passeth through F, which would be 4 a clock in the morning For if the line C be 9, the line D is 10, the next line is 11, and so forward till your account fall upon F: which must be 4 a clock past( 12 or) Midnight. Example 2. Upon the 8 of August, the star Aquila was seen on the Westsi de of the Meridian, and the hour of it was found, 3 and ½ an hour and ½ a quarter. The thread therefore being laid upon that star would be as the line A G, and the Bead( rectified to the ½ hour and ½ quarter) would stand at the point G. So that the next houreline on the left hand of G, must be called 3 a clock: and the line F must be 8 a clock. Then, the thread being removed to the day of your month ( August 8, in the upper circular line of moneths) will lie in the line A B: and the Bead at B will show the hour of the Night( if you keep your former account) to be ¼ and half past 1 a clock. For if F be 8 a clock( as is before expressed) then the last hour of the limb is 11, the first is 12, the second 1; beyond which, the Bead B is about 22 minutes of an hour. Therefore the hour of the night is 1 a clock 22 minutes. By these examples the manner of the work will sufficiently appear in all cases. The use of the Altimetrie Scale. THat Scale on the fore-side of the Quadrant next to the equal limb is here called the Altimerie Scale. It is numbered by 1, 2, 3, &c, to 10, 20, 30, &c, to 100. Each of which numbers are best supposed to be 100 fold. viz, 100, 200, &c, to 1000, 2000, &c: to 10000: and all the lesser parts estimated accordingly.— The ground on which you stand to make your mensuration, is also supposed to be a just level. I. To find any height at one observation. LEt your station be at E; and the sights D A directed to the point F: the thread A B cuts off the parts C B in the measuring Scale: which parts must be remembered.— Then measure from your station E, to the point H, which is just under F. And( always in this case( multiply this distance E H by the forenamed parts of C B, and from the product cut off 3 figures toward the right hand. The remainder is the Altitude G F. To which you must add H G, or D E, the height from your eye at D to your foot at E. Thus if the thread A B should cut off C B 1500 parts, and the distance E were 59 feet, The height G F would be 88. 500, or 88½ feet. II. To find part of an Altitude. LEt the length of F X be onely required. Standing then at E, you may find the altitude G F. Keep still the same standing at E, and find the altitude G X, by the last precedent. So G F taken from G X, gives F X required. III. Standing upon a known height, to find a Distance. LEt the height F H be known, and the distance H K be required. Order your standing so, that the two sights P, S; the point F, and the distance K, may all appear in one diagram right line. Then look what degrees the plummet cuts off in the equal limb from Q. Count the same number in the same limb, from S; and there lay the thread, as P T. Note then, what parts it cuts upon the measuring Scale from Q to T. Multiply those parts into F H the known Altitude: and from the product cut off three figures, the remainder or quotient, is the distance H K. Thus, if the thread P R should cut off Q R in the equal limb 56⅓ degrees, the same counted the other way, from S to T in the equal limb, and the thread laid thereto, would give 667 in the measuring Scale. Then F G being 88½ feet, and G H( suppose) five feet, F H must be 93½ feet. This multiplied into 667, makes 62364: from whence cutting away the three right hand figures, there remaines 62. 364 or 62⅓ feet, for the distance H K. IV. To find part of a distance. IF the distance of K from Z were required. First, find H K, then H Z, by the third precedent: their difference is K Z. If K Z were a trench, you might from the tower F, find the breadth of it without any approach unto it. V. To find a height at two observations. IF F H were to be measured, and the way from E to H were unpasseable, so that the distance of E from H could not be measured. You must in this case make two observations. For which purpose, Take your first station at E, and direct the sights D, A, to the point F: noting what parts the thread cuts upon the equal limb from C to B. Then go backward in a right line, to a competent distance, as to M; and there making a second station, observe( as before) what degrees the thread cuts upon the equal limb from N to O:( the two sights L, I, being justly directed to the point F). Then count these two arkes in the equal limb, from the contrary side of the quadrant, namely, from D to Y, and from L to u, and applying the thread thereto, look what parts it cuts from the measuring Scale at Y, and V. Take the lesser number of parts, out of the greater, noting the Difference. Measure also the distance of your two stations, namely from E to M, and add three ciphers to that measure. This last number must( in this kind of work) be divided always, by the fore-noted difference: and the quotient will give the Altitude of F above G. Example. Let the first observation cut off 38⅔ gr. in the equal limb. The second 56⅓ gr. Count the first ark from D to Y: the thread there laid gives 1250 in the measuring Scale. The second so counted from L to V, gives 667. The difference of these two, is 583. Let the distance of the stations measured from E to M, be 5160 feet. This number, with three ciphers added, is 5160000. Which divided by 583( the former difference) gives in the quotient, 8850 or 88½ feet for the height G F. And if G H be 5 foot more; The whole height H F will be 93½ feet. ¶ Note that in these mensurations, the point G is supposed to stand in the same level with the corner of your Quadrant D and L. So that G H, D E, L M, are all of one height. And note too, that the two stationary points are E and M, namely those which are just under the corners D and L. FINIS. Errata. Pag. Line. red. 6 1 Ninetyeth. 11 21 the thread laid at K. 14 25 horizon, is 29 gr. 21 min. 16 19·20 cuts from P. Count 25 13 Count the plains 26 19 the Suns altitude. 34 2 the line C D E 52 16 quadrant, close by 53 15 the same name or number 53 25 l⅓ part of a quarter. 56 1 Altimetric 56 4 Altimetric.