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<DATE>2015-01-27</DATE>
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<NAME>Alexis Jakobson</NAME>
<RESP>MURP</RESP>
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<BIBNO T="umi">99867886</BIBNO>
<VID>169725</VID>
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<TEXT LANG="eng">
<FRONT>
<DIV1 TYPE="title page">
<PB REF="1" MS="y">
<P>The u&s;es of a QUADRANT FITTED For daily practi&s;e.</P>
<P>Both with the ordinary lines for the <HI>Hour</HI> and <HI>Azimuth,</HI> and other things of the Suns cour&s;e in reference to the <HI>Horizon.</HI></P>
<P>And al&s;o with new Lines &s;erving to the fore-mentioned and other pur|po&s;es more accurately.</P>
<P><HI>As namely,</HI></P>
<P>To find the hour of the Night by the Stars;</P>
<P>To de&s;cribe the mo&s;t u&s;uall &s;orts of Dials;</P>
<P>To perform all common things in Men&s;uration.</P>
<P>And many other requi&s;ite conclu&s;ions.</P>
<P>Performed <HI>In an accurate, ea&s;ie, and delightfull way.</HI></P>
<P>By <HI>SAM. FOSTER,</HI> Profe&s;&s;or of A&s;tronomie in <HI>Gre&s;ham Colledg.</HI></P>
<P>PUBLISHED by A: T.</P>
<P><HI>LONDON:</HI> Printed for <HI>Francis Eglerfeild,</HI> and are to be &s;old at the Marygold in <HI>Pauls-Church-Yard;</HI> and by <HI>A: Thomp&s;on</HI> in <HI>Ho&s;ier-lane,</HI> neer <HI>Smithfield.</HI> 1652.</P>
</DIV1>
<DIV1 TYPE="to the reader">
<PB REF="2">
<PB REF="2">
<HEAD>AN Adverti&s;ement to the READER.</HEAD>
<P>_COncerning the &s;tructure and u&s;e of this Quadrant, the Reader may understand thus much. The Houre and Azimuth lines are like tho&s;e that are common|ly &s;een upon other Quadrants, and the u&s;es are (mo&s;t part) the &s;ame, and there|fore are lightly pa&s;&s;ed over; as is &s;een in the &s;e|cond Propo&s;ition. But the Distance of the Equi|noctiall and Tropiques is here &s;hortned, that &s;o more room might be gained above, for the better placing, and the more accurate dividing of the Equinoctials; which in &s;mall Instruments may receive each &s;econd degree, in larger each &s;ingle degree or more.</P>
<P>If it be required to make the&s;e yet larger,
<PB REF="3"> then may the fore-mentioned Azimuth lines be left quite out. For the u&s;e of them, as they are here de&s;cribed, is of &s;mall moment, very hardly, making good the Suns coa&s;t to one entire degree: and for &s;erious practi&s;e, the new Lines added are farre more &s;ufficient. If this be granted, then may the Equinoctials &s;tand belowe, by which means they &s;hall become large enough, even in &s;mall In&s;truments. E&s;pecially this may most fair|ly be done, if the hour-lines be reverted by chan|ging the places of the Equinoctiall and Tro|piques; that is, if the Equinoctiall altitudes be in&s;erted belowe on the circle neere&s;t the limb, and the Tropicall altitudes above in the circle neere&s;t to the center. Thus becoming more large, they will &s;upply all intended purpo&s;es very well. There is no Scheme given of this change now mentioned, nor of the vulgar Hours and Azi|muths, becau&s;e tho&s;e lines are well enough known alreadie, and this mutation is ea&s;ie to be under|&s;tood. If other Quadrants were thought com|pleat in u&s;e, this will be found much more copi|ous. For it &s;ervetb not onely to finde the hour of the day by the Sun: of the night by the Starrs, and what el&s;e belongs to their Ri&s;ings, Settings, Amplitudes, &amp;c. but is very well fitted al&s;o to de&s;cribe all the mo&s;t u&s;ual &s;orts of &s;tanding Dyals;
<PB REF="3"> that is, all that are upright, or el&s;e reclining or inclining to be full East and We&s;t: which two &s;orts will furni&s;h many kinds of &s;uch bo|dies, as are regularly formed. The&s;e are here performed by very ea&s;ie and familiar wayes of working. The nocturnall for the hour by the Stars, is more expedient in this then in other Quadrants. For in judging of time only by the Appul&s;e of the Stars to the Meridian, and finding that Meridian too onely by a rude con|jecture from the North Starr, an errour of a quarter or half an hour is ea&s;ily unawares com|mitted. This cannot be &s;o here if any ordina|ry care be had in taking the Starrs altitude. For this purpo&s;e, there are twelve &s;elect Stars in&s;erted, all of them of North Declination, ly|ing between the Equinoctiall and tropique of <HI>Cancer;</HI> and in &s;uch difference of right a&s;|cen&s;ions, as that one or other of them will be alwayes in &s;uch convenient place of the Heavens, <GAP DESC="illegible" RESP="KEYERS" EXTENT="2 letters"> from whence the hour may very fully be col|lected every night throughout the whole year. Since therefore they are &s;o convenient for u&s;e, there would be a little the more diligence u&s;ed to come to the knowledg of them in the Hea|vens, that due ob&s;ervations may be made when&s;oever any of them &s;hall be in view. If
<PB REF="4"> any de&s;ire that other &s;tarrs (&s;uch as are better known to them) &s;hould be in&s;erted, they may have their de&s;ire ea&s;ily fulfilled. Onely they must take care, that the &s;tarrs be &s;uch as fall between the Tropiques and the Heavens, and chiefly between the Equin<GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">ctial and North Tro|pique; becau&s;e &s;uch &s;tars are longe&s;t in view, and their hours be&s;t found&mdash;The Propo&s;itions that are here &s;et down might have been increa&s;ed both in number and in variety of performance, if pro|lixity had been affected; but &s;uch of them, and &s;uch wayes of effecting them, are here pitched up|on as &s;eemed most conducible for daily u&s;e, and to make the Treati&s;e, rather materiall&punc; then burthen&s;ome. And for the &s;ame rea&s;on it is, that the &s;everall lines upon the Quadrant, are denoted by letters only; that by &s;uch bre|vity, all unnece&s;&s;ary circumlocution might be taken off, which, by impo&s;ition of names to each of them, could not &s;o ea&s;ily have been avoided. If other Quadrants have hereto|fore found good acceptance, becau&s;e they were of &s;ome good u&s;e, I &s;hould in rea&s;on expect a greater proportion of thanks from the ingenious for making publique this larger Improvement of this In&s;trument But be that as the <HI>READER</HI> li&s;teth.</P>
<PB REF="4">
<P> The&s;e additionall lines were invented, and the u&s;es written at the reque&s;t of a Friend, and are intended as an addition or Ap|pendix unto Master <HI>Gunters</HI> Quadrant, it being most proper for the &s;ame, onely &s;ome few are publi&s;hed alone in this volumn, for the private &s;atisfaction of &s;ome Friends, of the Judicious Author Mr. <HI>SAMUEL FOSTER.</HI></P>
<CLOSER><SIGNED><HI>Thine,</HI> A: Thomp&s;on.</SIGNED></CLOSER>
</DIV1>
<DIV1 TYPE="illustration">
<PB REF="5" MS="y">
<P><FIGURE>
<FIGDESC>depiction of a quadrant</FIGDESC>
</FIGURE></P>
</DIV1>
</FRONT>
<BODY>
<DIV1 TYPE="text">
<PB N="1" REF="5">
<HEAD>The u&s;es of the QVADRANT.</HEAD>
<DIV2 N="1" TYPE="part">
<HEAD>I. <HI>To finde the Suns Declination.</HI></HEAD>
<P>_LAy the threed to the day of the moneth upon the back &s;ide of the Quadrant, and it will &s;hew you the Declination of the Sun in that unequall &s;cale, which is numbred with twice 23&frac12;. If your day fall in the upper &s;cale of moneths (which may be cal|led the Summer &s;cale) then is the declination North: if it fall in the lower (or Winter) &s;cale, the declination is South from the Equi|noctiall.</P>
<P>Thus upon <HI>Aprill</HI> 20, you &s;hall find the &s;un to decline 15 <HI>gr.</HI> northward: and <HI>January</HI> 30, it declines about 14&frac12; <HI>gr.</HI> &s;outhward.</P>
<P>&para; The contrary work is ea&s;ie: by a&s;&s;ign|ing the &s;uns declination, to know on what day of the moneth the &s;ame &s;hall be. For the
<PB N="2" REF="6"> threed may be laid to the declination in two places; in both which it will cro&s;&s;e the two halfe years, &s;hewing two &s;everall dayes on which the &s;un &s;hall have &s;o much declination North; and two more dayes on which it &s;hall have that declination Southward. It will be ea&s;ie to di&s;tingui&s;h which of the&s;e days &s;erves your purpo&s;e, by the two &s;ea&s;ons of the year unto which the two &s;cales of moneths doe an&s;wer.</P>
</DIV2>
<DIV2 N="2" TYPE="part">
<HEAD>II. <HI>To rectifie the Bead for ob&s;ervation of Houre or Azimuth: and to perform tho&s;e things that are done by the u&s;uall lines upon the Quadrant.</HI></HEAD>
<P>HAving found the Suns declination for your day, you mu&s;t count the &s;ame upon the double equall &s;cale which is on the fore&s;ide 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 threed being laid thereto, you mu&s;t move the bead, til it fal ju&s;tly upon the houre of 12, &s;o &s;hall it be &s;et right for the intended u&s;es of that day. As,</P>
<P N="1">1 <HI>For the</HI> HOURE. If you ob&s;erve the &s;uns
<PB N="3" REF="6"> altitude (by letting the Sun-beames to &s;hine through the &s;ights, and the Plummet to hang at full liberty, clo&s;e to the plane of your qua|drant) the bead will &s;hew the houre, if you have re&s;pect to the time of the yeer. That is; If the &s;uns declination be North, the bead &s;hewes the time of the day among the Sum|mer houres, tho&s;e which &s;pread from the Equinoctiall towards the right hand. If the Sun decline South, the time mu&s;t be accounted in the cro&s;&s;e lines which are the winter hours. And in this ob&s;ervation you &s;hall &s;ee the threed to cut (in the equall limbe) the Suns altitude above the Horizon.&mdash;Thus at <HI>Lon|don,</HI> if the &Sun; decline 15 <HI>gr.</HI> Northward, and the altitude were 95/12 <HI>gr.</HI> the hour would be about a quarter before 6 in the morning: Or a quarter pa&s;t 6 in the evening. But if the Sun had the &s;ame declination Southward, and the &s;ame altitude al&s;o, then would the time be half an hour pa&s;t 8, in the morning; Or halfe an hour pa&s;t 3, in the evening. The former of the&s;e times is &s;hewed by the bead among the &s;ummer-houres; the latter among the winter-hours.</P>
<P N="2">2 <HI>For the</HI> AZIMUTH. If the &s;uns altitude be numbred the contrary way in the equall limbe, and the threed be laid thereto, the bead
<PB N="4" REF="7"> will then &s;hew the <HI>Azimuth</HI> of the Sun if you account it according to the time of the yeer: That is, among the Summer <HI>Azimuth;</HI> when the &s;un hath North declination; and among the Winter Azimuths when the &s;un declines South. The Summer Azimuths, are tho&s;e that &s;pread from the Equinoctiall towards the left hand: The other cro&s;&s;ing them, are the Win|ter Azimuths.&mdash;Thus if the &s;uns declination were 8 <HI>gr.</HI> Northward, and the altitude 18 <HI>gr.</HI> the Azimuth would be 80 <HI>gr.</HI> from the South. But if the &s;un had 8 <HI>gr.</HI> of South declination, and 18 <HI>gr.</HI> altitude, the Azimuth would be 50 <HI>gr.</HI> from the South, here at <HI>London.</HI> This way may &s;erve for gro&s;&s;e workes, when the Azimuth is required onely within one or two whole degrees. You &s;hall finde it done more accurately and for better purpo&s;es in the thirteenth follow|ing.</P>
<P N="3">3 <HI>For the</HI> ASCENSIONALL <HI>difference.</HI> The bead being (rectified as before) and ap|plied to the left &s;ide of the quadrant, gives the A&s;cen&s;ionall difference, or the time of &s;un-ri|&s;ing &amp; &s;eting before or after 6 a clock, among tho&s;e hours and quarters which inter&s;ect each other upon the &s;ame left &s;ide of the quadrant, if you account them agreeable to the time of
<PB N="5" REF="7"> the yeer. And from the bead to the line of 12 rightly taken according to your time of Sum|mer or Winter, gives the &s;emidiurnall Ark of the &s;un, or half the dayes lenght.&mdash;As al&s;o, from the bead to the other line of 12, which &s;erves for the contrary time of the yeer, gives the &s;eminocturall Ark, or half the length of the night.&mdash;Thus if the &s;uns declina|tion were 14&frac13; <HI>gr.</HI> the A&s;cen&s;ionall difference would be 1 hour and &frac14; of an houre. And if the &s;aid declination were North, then the &s;un ri&s;eth that day &frac14; of an hour before 5; &s;etteth &frac14; after 7: The &s;emidiurnall arke (from the bead to the Summer 12) is 7&frac14; hours. The &s;eminocturall arke (from the bead to the Winter 12 is 4&frac34; houres. The&s;e doubled, make the day 14&frac12; hours long, The night 9&frac12; long.</P>
<P N="4">4 <HI>For the</HI> AMPLITUDE. The bead ap|plyed to the right &s;ide of the quadrant gives the <HI>Amplitude</HI> of &s;un ri&s;ing and &s;etting, in all varieties. Namely; From the bead to that South Azimuth which is proper to the &s;ea&s;on of the yeer, is the <HI>Amplitude</HI> from &s;outh: as al&s;o to the contrary &s;outh Azimuth, gives the <HI>Amplitude</HI> from North: &s;hewing how many degrees of the Horizon the &s;un ri&s;eth and &s;et|teth any day from the ju&s;t South or North. So from the bead to the Ea&s;t and We&s;t Azi|muth
<PB N="6" REF="8"> (which is the ninth Azimuth) gives the Amplitude from Ea&s;t or We&s;t.&mdash;Thus if the &Sun; decline 14&frac13; <HI>gr.</HI> The Amplitude is here, 23&frac12; <HI>gr.</HI> almo&s;t. If the declination be North, then is this Amplitude from Ea&s;t and We&s;t toward the North, 23&frac12; degrees. The Amplitude from the north it &s;elf is then 66&frac12; <HI>gr.</HI> From the &s;outh point of the Horizon it is 113&frac12; <HI>gr.</HI> You may ea&s;ily (in &s;uch manner) ac|count it for South declinations of the &Sun;.</P>
</DIV2>
<DIV2 TYPE="part">
<HEAD>V. <HI>To finde when Twilight begins in the Morn|ing and ends at Evening: which moments are the two utmo&s;t termes of darke night.</HI></HEAD>
<P>AFter the bead is rectified for your day, The threed laid to 18 <HI>gr.</HI> in the equall limbe, will &s;hew the houre or part required. Onely here remember to take your hour a|right. Namely, in winter time, look among the Summer houres, where it is that the bead re&s;t|eth, for that is the Morning or Evening hour of Twilight. So in &s;ummer time you mu&s;t look among the winter hours.&mdash;Thus when the &Sun; declines 111/10 <HI>gr.</HI> Southward, the twi|light begins at <HI>London</HI> at 5 in the morning, and ends at 7 a clocke at night, as the bead
<PB N="7" REF="8"> &s;hewes among the &s;ummer houres. But if that declination were North, the twilight would begin at &frac14; of an hour before 3 in the morning, and ende &frac14; after 9 at night.&mdash;The &s;uns depre&s;|&s;ion 18 <HI>gr.</HI> under the Horizon, is the u&s;uall terme whereon to begin and end the twilight. You may as well do this to any degree of light, as to 12, or 13 degrees depre&s;&s;ion: At which time in the morning all things begin to be vi&s;ible, and the light to be of &s;ome u&s;e. As if the &Sun; decline 3&frac12; <HI>gr.</HI> Southward, if you &s;et the bead thereto, and then lay the threed at 12 <HI>gr.</HI> in the equall limbe, you &s;hall &s;ee the bead (among the &s;ummer houres,) fall upon 5, in the morning and 7, at right, So that at 5, and till 7, there is a rea&s;onable degree of light. Or if in &s;ummer the &Sun; had declined 7&frac12; <HI>gr.</HI> Northward; the &s;aid degree of light would begin at 4, in the morning, and end at 8 in the evening.&mdash;Neer to the longe&s;t days you &s;hall finde no twilight at all according to 18 degrees depre&s;&s;ion of &Sun; under the Hori|zon: for then the bead will fall be the Winter 12 a clock-line.</P>
<P>&para; The&s;e are the chief u&s;es of the hour and Azimuth lines as they are here, and in all Quadrants commonly in&s;erted. There are other things concerning the &s;uns place in
<PB N="8" REF="9"> the Ecliptick: The &s;uns declination: The &s;uns right a&s;cen&s;ion: Namely,&mdash;How by having any one of the&s;e, to finde out the re&s;t&mdash;The&s;e are here omitted as matters onely of curio&s;i|ty, being of no further u&s;e, in this in&s;trument, then that they may be known. Yet if any &s;hould de&s;ire them, they may have a Scale of the 12; &s;ignes in&s;cribed on the back &s;ide; by help of which, the fore-named requi&s;ites may be attained. The particulars that follow are mo&s;t aimed at, (as being more of them, and more accurate) and therefore the precedent things are thus briefly pa&s;&s;ed over.</P>
</DIV2>
<DIV2 N="3" TYPE="part">
<HEAD>III. <HI>To finde the Suns A&s;cen&s;ionall difference, &amp;c.</HI></HEAD>
<P>COunt the declination in the equall limbe from F. to K. The threed there laid, gives BS the A&s;cen&s;ionall diffe|ence.&mdash;The &s;aid a&s;cen&s;ional difference gives the times of &s;un ri&s;ing and fetting be|fore and after 6: with the lengths of Day and Night.&mdash;The &s;ame may be done for all &s;tars who&s;e declinations are known.</P>
<P>&para; So by having the a&s;cen&s;ionall difference,
<PB N="9" REF="9"> you may finde the &s;uns declination thereto belonging.</P>
<P>Here at <HI>London,</HI> if the declination be 20 <HI>gr.</HI> the a&s;cen&s;ionall difference is 27 <HI>gr.</HI> 14 min. That is, 1 houre, 49 minutes. And if this declination be North, the &s;un ri&s;eth 1 hour 49 min. before 6, and &s;etteth &s;o much after 6. That is, it ri&s;eth 11 min. after 4 in the morn|ing; and &s;etteth 49 min. after 7 a clock at night. And the time of &s;etting being doubled, gives 15 hours, 38 min. for the dayes length. The time of ri&s;ing being doubled, gives 8 hours, 22 min. for the length of night. But if the declination had been South, the &s;un &s;hould ri&s;e 1 hour, 49 min. after 6 (that is, at 7, 49 min.) And &s;hould &s;et 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.</P>
</DIV2>
<DIV2 N="4" TYPE="part">
<HEAD>IV. <HI>To finde the &s;uns Amplitude, &amp;c.</HI></HEAD>
<P>COunt the declination in the equall limbe from G to H. The threed there laid, gives C R for the Amplitude.&mdash;The &s;ame may be done for &s;tars who&s;e declinations are known.</P>
<PB N="10" REF="10">
<P> &para; So by having the Amplitude, you may finde the declination. For if the Amplitude be counted from C to R, the threed laid at R, gives the declination G H.</P>
<P>At <HI>London,</HI> if the declination be 20 <HI>gr.</HI> the Amplitude is 33 <HI>gr.</HI> 20 min. from the Ea&s;t and We&s;t points of the Horizon.</P>
</DIV2>
<DIV2 N="5" TYPE="part">
<HEAD>V. <HI>Having the declination of any upright plane, to finde the elevation of the &s;tyle, &amp;c.</HI></HEAD>
<P>LAy the threed to the planes declination, counted from D to R: So will G H be the Elevation.</P>
<P>&para; So by having the Elevation G H, you may finde D R the declination.</P>
<P>If an upright plane (here) decline 20 <HI>gr.</HI> the &s;tyles elevation will be 35 <HI>gr.</HI> 48 minutes</P>
</DIV2>
<DIV2 N="6" TYPE="part">
<HEAD>VI. <HI>To finde the Deflexion, &amp;c.</HI></HEAD>
<P>COunt the declination from B to S. The threed there laid, gives F K the Deflex|ion.</P>
<P>&para; So by having F K the Deflexion, you may finde B S, the planes declination.</P>
<PB N="11" REF="10">
<P> If A plane declining 20 <HI>gr.</HI> the Deflex|ion is 15 <HI>gr.</HI> 13 min.</P>
</DIV2>
<DIV2 N="7" TYPE="part">
<HEAD>VII. <HI>To finde the planes Difference of longitude, &amp;c.</HI></HEAD>
<LIST>
<ITEM>1 COunt the Elevation from F to K. E S is the difference of longitude.</ITEM>
<ITEM>2 Count the Deflexion from G to H. C R is the difference of longitude.</ITEM>
</LIST>
<P>&para; By the contrary works, having the difference of longitude, you may finde the Elevation and Deflexion.</P>
<P>A plane declining 20 <HI>gr.</HI> hath 25 <HI>gr.</HI> diff. of longitude.</P>
</DIV2>
<DIV2 N="8" TYPE="part">
<HEAD>VIII. <HI>To make an Horizontall Diall.</HI></HEAD>
<P N="1">1. COunt the houre from E to S: The threed laid at S, gives F K. Then count G H equall to F K: The threed there laid gives D R, the &s;pace of that hour from 12.</P>
<P N="2">2. Count the hour from C to R, and by help of the threed you &s;hall have G H. Then count FK equall to GH: the threed laid at R, gives B S, for the &s;pace of that hour from 12.</P>
<P N="3"><PB N="12" REF="11"> 3 With a paire of compa&s;&s;es, take the hour from C to R, and &s;et it from B to S. B S is the &s;pace or angle of that houre from twelve.</P>
<P N="4">4 Take with your compa&s;&s;es the hour from E to S, and &s;et it from D to R. So the number D R &s;hews how many degrees that houre mu&s;t be from 12.</P>
<P>By all the&s;e waies (here at <HI>London)</HI> the third hour will be found about 38 <HI>gr.</HI> from 12. The re&s;t will be in like manner found according to their true quantities.</P>
</DIV2>
<DIV2 N="9" TYPE="part">
<HEAD>IX. <HI>To finde what Angle any hour-circle ma|keth with the Horizon; or any Azimuth makes with the Equinoctiall.</HI></HEAD>
<P>LEt the number of the hour-circle (or <HI>Azimuth)</HI> from &s;outh, be counted from C to R the threed laid at R, will cut the equall limbe in H. And F H will be the an|gle required.</P>
<P>&para; By the angle know, it will be ea&s;ie by
<PB N="13" REF="11"> the contrary worke, to finde the houre (or Azimuth) to which that angle be|longeth.</P>
<P>The third houre (or 45 Azimuth) makes with the Horizon (or with the Equinoctiall) an angle of 63 <HI>gr.</HI> 53 <HI>minutes</HI> here at <HI>Lon|don.</HI></P>
</DIV2>
<DIV2 N="10" TYPE="part">
<HEAD>X. <HI>To finde what arke of any houre-circle is in|tercepted betweene the Equinoctiaell (or any Parallel) and the Horizon.</HI></HEAD>
<P>COunt the number of the houre-circle from South, from E to S: or if it be a|bove 90, from E to B, and back again to S. So F K in the equall limbe, will be the arke required, betweene the Equinoctiall and Ho|rizon.</P>
<P>The arke intercepted between any parallel and the Horizon, may hence al&s;o be found.&mdash;If the Declination of the parallel be North, and the houre be between 12 and 6, Add the declination to the arke found by the former worke: In other houres beyond 6, &s;ub|tract the former ark out of the declination, the re&s;ult will be the ark required. Upon the hour
<PB N="14" REF="12"> of 6 it &s;elf, the declination of the parallel is the ark intercepted.&mdash;If the declination be South, Subtract it out of the arke found before, (namely, the ark intercepted between the Equinoctiall and Horizon) what remains is the ark intercepted between that parallel and the Horizon.</P>
<P>Thus at <HI>London.</HI> The ark of the 3 hour inter|cepted between the Equinoctiall and Hori|zon, is 29 <HI>gr.</HI> 21 min.&mdash;And if the decli|nation be 18 <HI>gr.</HI> North, the arke intercepted between that parallel and the horizon, is 47<HI>gr.</HI> 21 min.&mdash;If the parallel be 18 <HI>gr.</HI> South, the arke will be 11 <HI>gr.</HI> 21 min.</P>
<P>&para; The fir&s;t worke, will al&s;o &s;hew what arke of any <HI>Azimuth</HI> from South is intercepted be|tweene the Horizon and Equinoctiall, if in &s;tead of the hour-circle from South, you u&s;e the <HI>Azimuth</HI> from South. This intercepted arke, is the Equinoctiall altitude of that <HI>A|zimuth.</HI></P>
<P>So in the 45 <HI>Azimuth</HI> from South, the Equinoctiall is 29 <HI>gr.</HI> 21 min. high. In the 135 <HI>Azimuth</HI> from South, the Equinoctialls de|pre&s;&s;ion under the horizon, in 29 <HI>gr.</HI> 21 min.</P>
<P>This is made u&s;e of afterwards.</P>
</DIV2>
<DIV2 N="11" TYPE="part">
<PB N="15" REF="12">
<HEAD>XI. <HI>How high the Sun &s;hall be upon any Azi|muth, and in any Declination.</HI></HEAD>
<P>THe <HI>Azimuth</HI> is be&s;t numbred from the &s;outh. And this propo&s;ition (with mo&s;t of tho&s;e that follow) is to be done by helpe of compa&s;&s;es.</P>
<P>&para; If the &Sun; be in the Equinoctiall, the fir&s;t worke of the la&s;t propo&s;ition gets the Equi|noctiall altitude or depre&s;&s;ion, by counting the <HI>Azimuth</HI> from E to S, whereby the arke F K will be found. This arke (if the <HI>Azi|muth</HI> be le&s;&s;e than 90) is the altitude: if more than 90, it is the depre&s;&s;ion.</P>
<P>But if the &s;un have Declination, then Fir&s;t, lay the threed from F towards K according to that Declination, and take the lea&s;t di&s;tance from the point B, to your threed, and keep this extent. Then,</P>
<P>&para; If the &s;uns declination be &s;outh; Count your <HI>Azimuth</HI> from E to S, and lay the threed 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 &s;ide A B, applying the threed to the remote&s;t di&s;tance of that circuit. The threed &s;o laid, will give the alti|tude
<PB N="16" REF="13"> required; if you count the degrees from F. Thus the &s;un declining &s;outh 11 <HI>gr.</HI> 30&prime; will have 16&frac12;<HI>gr.</HI> of altitude in the 45 <HI>Azi|muth.</HI></P>
<P>&para; If the &s;uns declination be North,&mdash;and the <HI>Azimuth</HI> le&s;&s;e then 90 from &s;outh; Count your <HI>Azimuth</HI> from E to S, and lay the threed at it, and let it cut E N, in T. Then &s;et one foot of your former extent in T, and with the other foot turned about, lay the threed at the remote&s;t di&s;tance, from T to|wards the &s;ide A C. The theed &s;o lying, &s;hews from F in the equall limbe, the altitude requi|red. Thus if the &s;un decline 11&frac12; <HI>gr.</HI> North, his altitude upon the 45 <HI>Azimuth</HI> will be 42&frac15; <HI>gr.&mdash;</HI>But if the Azimuth be more than 90, count from B to S, the exce&s;&s;e above 90; and applying the thread thereto, &s;ee what de|grees of the equall limbe the threed cuts from From F. Count that number of de|grees from 60 (in the equal limbe) forwards, toward 70, 80, 90, and lay the threed there, which &s;uppo&s;e to cut the line &agr; &ohgr;, in &pgr;. Set your compa&s;&s;e (keeping &s;till their fir&s;t extent) upon &pgr;, and turn the other foot towards the the &s;ide A C, laying the treed at the remote&s;t turne. If now to the threed &s;o laid, you num|ber the degrees in the equall limbe, from
<PB N="17" REF="13"> 60, the &s;ame &s;hall be the altitude required. Thus if the &s;un decline 11&frac12; <HI>gr.</HI> North, and the Azimuth be 101&frac14; <HI>gr.</HI> from South, the al|titude mu&s;t be 5&frac34;<HI>gr.</HI> in our latitude of 51 <HI>gr.</HI> 30 minutes.</P>
<DIV3 TYPE="section">
<HEAD>Another way for this Propo&s;ition.</HEAD>
<P>BY the fir&s;t work in this 11^t^h, get the Equi|noctial altitude or depre&s;&s;ion for your Azi|muth. Then lay the threed at E: and in C D, from D, count the &s;aid altitude or depre&s;&s;ion; from which number or point, take the lea&s;t di&s;tance to the &s;ide A C. Enter this length between the &s;ide A C and the threed, keeping one foot upon the line A C, remove it there|on to and fro, till the other foot turned about may ju&s;tly touch the threed. Then keeping your compa&s;&s;es there &s;et, remove the threed from Gtoward H, according to the Suns declination, and take the lea&s;t di&s;tance from your former &s;tanding to the threed. This length mea&s;ured in the Scale CD [&s;o as one foot &s;tanding upon the Scale, the other turned about may ju&s;t|ly touch the &s;ide A C] &s;hewes an arke, Which,
<LIST>
<PB N="18" REF="14">
<ITEM>If the &s;uns Declination be &s;outh, mu&s;t be &s;ub&s;tact|ed from
<LIST>
<ITEM>the Azimuths Equinoctiall Al|titude.</ITEM>
</LIST></ITEM>
<ITEM>If the &s;uns Declination be north, and the Azimuth le&s;&s;e than 90, mu&s;t be added to</ITEM>
</LIST></P>
<P>If the &s;uns Declination be north, and the Azimuth more than 90, the Azimuths Equinoctiall depre&s;&s;ion mu&s;t be taken out of this arke;</P>
<P>The re&s;ult is the altitude looked for.</P>
<P>Thus if the Azimuth be 70/110 from &s;outh, the Equinoctiall Altitude / Depre&s;&s;ion will be 15&frac14; <HI>gr.</HI></P>
<P>The arke found, will be 14&frac14;. Then, If the &Sun; decline 11&frac12; &s;outh, the altitude up|on the 70 Azimuth wil be 1 degree.</P>
<P>If the &Sun; decline 11&frac12; north, the Altitude upon the 70 Azimuth will be 29&frac12; degrees.</P>
<PB N="19" REF="14">
<P> If the &s;uns declination were 20 <HI>gr.</HI> north, That forementioned arke would be 25 <HI>gr.</HI> whence taking 15&frac14;, there re|maines 9&frac34; for the Altitude of the &s;un upon the 110 Azimuth from &s;outh, at that Declination of 20 <HI>gr.</HI> north.</P>
<P>&para;. By this worke, may a table of altitudes be made, by which the former Azimuth lines upon the quadrant may be in&s;erted.</P>
</DIV3>
</DIV2>
<DIV2 N="12" TYPE="part">
<HEAD>XII. <HI>To finde how high the Sun &s;hall be at any houre, and in any Declination.</HI></HEAD>
<P>FIr&s;t, finde the intercepted arke of your houre, betweene the parallel of Declina|tion, and the Horizon, by the tenth.</P>
<P>Secondly, finde what angle your hour-cir|cle maketh with the Horizon, by the ninth.</P>
<P>Thirdly, count that angle from C towards D, and from thence take the lea&s;t di&s;tance to the &s;ide A C. Mea&s;ure this length upon the &s;ide A C (from A) and there &s;et your com|pa&s;&s;es. Then keeping that &s;tation of your com|pa&s;&s;es, Lay the threed to the intercepted ark, counted in the equal limbe from G: and take the lea&s;t di&s;tance from your &s;tanding to the threed. Set one foot of this length in the
<PB N="20" REF="15"> &s;cale C D, &s;o, as that the other being turned about, may touch the &s;ide A C, &s;o &s;hall that foot in the &s;cale C D, give the Degrees of Al|titude required, if you number them from C.</P>
<P>Let the houre be three from Ncon. The intercepted arke between the Equinoctiall and Horizon, will be 29 <HI>gr.</HI> 22 minutes. And if the Sun decline North / South 11&frac12; <HI>gr.</HI> The intercept|ed arkes will be 40.52/17.52. And the angle of the third houre with the Horizon, is 63 <HI>gr.</HI> 53&punc; So that the Altitude for North / South Declination of 11&frac12; <HI>gr.</HI> will be 36/16 degrees.</P>
<P>&para; By this work you may make a Table of the Suns altitudes upon any parallel of De|clination. And by tho&s;e altitudes you may in|&s;ert tho&s;e Summer and Winter houres which are upon the Quadrant.</P>
</DIV2>
<DIV2 N="13" TYPE="part">
<HEAD>XIII. <HI>To finde the Suns Azimuth.</HI></HEAD>
<P>FIr&s;t, lay the threed to the Suns Declinati|on counted in the equal limbe from F to K, and take the lea&s;t di&s;tance from the point B,
<PB N="21" REF="15"> to the threed, and keep your compa&s;&s;es at that extent. Then count the Suns Altitude in the equall limbe, from F, and lay the threed to it. This being done,</P>
<P>&para; If the Sun Decline South, keep one foot of your compa&s;&s;es always upon the line E N, beyond the threed, towards E, and re|move it &s;till upon that line, till the other foot being turned about may touch the threed preci&s;ely. Ob&s;erve then, where the foot of your compa&s;&s;e &s;tandeth upon the line EN &s;uppo&s;e at V. Bring the threed to V, and it &s;hewes (from E) the Azimuth from the South.</P>
<P>&para; If the Sun Decline North, keep one foot of your former extent, upon the line E N, on this &s;ide the threed, towards N, and remove it &s;till upon that line, until the foot that is turned about do touch upon the threed. And ob|&s;erve where your compa&s;&s;e foot then &s;tand|eth, upon the line E N (&s;uppo&s;e it &s;tand at W.) Lay the threed at W, and it will cut the &s;cale E B. The parts whereof, from E to the threed, are the Azimuth from South.</P>
<P>But if it &s;o fall out in North Declinations, that when the threed is laid to the altitude, you cannot finde roome upon the line E N, whereon to &s;et your compa&s;&s;es &s;o as to keep
<PB N="22" REF="16"> the conditions before required; then work in this manner. Add alwayes 30 degrees to the Suns altitude, and lay the threed at that compound Altitude, numbred in the equall limbe from F. To the threed &s;o laid, enter the former extent of your compa&s;&s;es, between the threed and the line &agr; &ohgr;, keeping one foot alwayes upon that line. And look where the foot of your compa&s;&s;es re&s;teth upon that line; &s;uppo&s;e at &pgr;. Take then the length from &pgr; to &agr;, and &s;et it upon the line N E (from N towards E): and to the point whete it re&s;ts, apply the threed: ob&s;erving what parts it cuts upon the &s;cale, from B. The number of tho&s;e parts, gives the quantity of the Azimuth above 90 from the South. Or the parts cut from E, give the Azimuth from the North.</P>
<P>&para; If the &Sun; decline not at all, but is in the Equinoctiall, then the &s;ole Altitude from F to K (by helpe of the threed thereto applied) gives E S the Azimuth from South.</P>
<P>If the Altitude of the &Sun; be 21&frac23; in the Equinoctiall, the Azimuth from South is 60 degrees.</P>
<P>If the Sun decline South 5 <HI>gr.</HI> and the Alti|tude were 15&frac34; <HI>gr.</HI> the Azimuth would be 60 degrees.</P>
<PB N="23" REF="16">
<P> If the &Sun; decline North <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">0 <HI>gr.</HI> and the Al|titude were 50, the Azimuth would be found 50 <HI>gr.</HI></P>
<P>If the &Sun; decline North 20 <HI>gr.</HI> and the Alti|tude were 9&frac34; <HI>gr.</HI> the Azimuth would be 110 <HI>gr.</HI> from the South.</P>
<P>&para; If you &s;uppo&s;e the &s;un to have no Alti|tude, and do work by the&s;e rules, you &s;hall finde the &s;uns Amplitude, Or|tive and Occa&s;ive, from the South. As if the &s;un Decline 20 <HI>gr.</HI> North, you will finde 123 <HI>gr.</HI> 20 <HI>min.</HI> for the Amplitude from the South.</P>
</DIV2>
<DIV2 N="14" TYPE="part">
<HEAD>XIIII. <HI>To finde the houre of the day, by the Sun.</HI></HEAD>
<P>COunt the &s;uns Altitude in the equall limbe from F: and to the threed there laid, take the lea&s;t di&s;tance from the point B: and keep this di&s;tance.</P>
<P>Then count the &s;uns Declination (which is had ea&s;ily by the fir&s;t propo&s;ition:) from F in the equall limbe, and apply the threed, to it. Then further,</P>
<P>&para; If the declination be South, &s;et one foot of your former extent, upon the line E N (al|wayes
<PB N="24" REF="17"> on that &s;ide the threed on which E &s;tandeth from it) and remove it thereon, till the other (turned about) may ju&s;tly touch the threed A K. Suppo&s;e (in &s;o doing) the compa&s;&s;e foot &s;tayeth at V. The threed ap|plied to the point V, will cut the houre from Noon, if you count the intercepted parts upon E B, from E.&mdash;Thus if the &s;un decline 20 degrees South, and the Altitude were 13 <HI>gr.</HI> 50 <HI>min.</HI> the houre at <HI>London</HI> would be 10, or 2.</P>
<P>&para; If the Declination be North, &s;et one foot of your former extent upon the &s;ide A C, removing it thereon to and fro, till the other foot turned about, will onely touch the threed. When it is &s;o fitted, let that foot upon the &s;ide A C, keep its &s;tation; and from thence extend the other foot to the &s;uns De|clination counted in the &s;cale A P This la&s;t extent mu&s;t be applied to the line N E, from N: and where it &s;tayes, lay the threed. So the parts cut upon the &s;cale E B, will give the houre.&mdash;But this mu&s;t be done with cau|tion For if that foot that kept its &s;tation, &s;tood from A, beyond the Suns Declination in the &s;cale A P, then the intercepted arke from E to the threed, gives the houre from Noon. But if the forenamed foot &s;tood be|tween
<PB N="25" REF="17"> A and the Declination, then the whole arke E B 90, with the arke from B back again to the threed (the&s;e two put together) give the hour from Noon.</P>
<P>Thus, if the &s;un decline 15 <HI>gr.</HI> Northward, and be 21 <HI>gr.</HI> high, the houre is 7 before, or 5 after, Noon. Or if the altitude were 2&frac23; <HI>gr.</HI> the houre mu&s;t have been 5 in the Morning, or 7 in the evening: namely, 90 and 15 de|grees from Noon.</P>
</DIV2>
<DIV2 N="15" TYPE="part">
<HEAD>XV. <HI>On an &V;pright declining plane, to finde the angle between</HI> 12 <HI>and</HI> 6.</HEAD>
<P>COut the palnes Declination from C towards D: From that point take the lea&s;t di&s;tance to the &s;ide CA. Set that length from M to Y, upon the line M Y. The threed laid at Y gives G K for the angle betweene 12 and 6.</P>
<P>Or count the Declination of the plane from B towards E, and lay the threed at it. The threed will cut N E. Take from N to the inter&s;ection, and apply it to M Y; the threed put to Y gives G. K, as before.</P>
<PB N="26" REF="18">
<P> If a plane Decline 20 <HI>gr.</HI> this angle will be 66&frac34; at <HI>London.</HI></P>
</DIV2>
<DIV2 N="16" TYPE="part">
<HEAD>XVI. <HI>To finde the Declination of a plane.</HI></HEAD>
<P>FIr&s;t, draw an Horizontall line upon your plane (which you may do by your qua|drant.) Then apply one &s;ide of the quadrant to that line, &s;o as the limbe may be toward the &s;un, and the plane of the quadrant may lye Horizontally flat. Thirdly, having a loo&s;e threed and plummet, you mu&s;t hold that threed clo&s;e by the edge of the limbe (letting the plummet hang down at liberty) till the &s;hadow of the threed pa&s;&s;eth directly through the quadrants center. Which done, you &s;hall &s;ee what degrees of the limbe the &s;hadow cuts from that &s;ide of the quadrant which is per|pendicular to the Horizontall line. This is called the Horizontall di&s;tance. At the &s;ame moment of time, ob&s;erve the &s;ame Altitude. By this Altitude you may get the &s;uns Azi|muth from South, by the thirteenth.</P>
<P>After this preparation, take diligent notice, whether the &s;hadow of the threed fall be|twixt the South and the perpendicular &s;ide of the quadrant, Or whether the &s;ame &s;hadow
<PB N="27" REF="18"> fall &s;o, as to leave both the South and the &s;aid perpendicular &s;ide (both of them) upon one coa&s;t of the &s;hadow.</P>
<P>In the fir&s;t ca&s;e, you mu&s;t add the Horizon|tall di&s;tance to the Azimuth. In the latter ca&s;e, you mu&s;t &s;ub&s;tract the le&s;&s;er out of the greater. The re&s;ult (whether it be &s;umme or difference) gives the planes Declination from the South.</P>
<P>Note here in the &s;econd ca&s;e. That if the Horizontall di&s;tance be greater then the A|zimuth, then doth the plane decline to that coa&s;t (Ea&s;t or We&s;t) which is contrary to the coa&s;t on which the &s;un &s;tood from the South. This falleth our very frequently.</P>
<P>Note al&s;o in the fir&s;t ca&s;e. That if the &s;umme of the Horizontall di&s;tance and Azi|muth do exceed 180 <HI>gr.</HI> then the planes De|clination from South, is contrary to that coa&s;t whreon the &s;un &s;tood. And it is found, by &s;ub&s;tracting the forementioned &s;umme out of 360 degrees. This happens more &s;el|dome: that is, onely upon &s;ome North planes; and on them, onely then, when the &s;uns Azi|muth is more than 90 from the South; and the Horizontall di&s;tance more than is the A|zimuth from the North.</P>
<P>Examples are here omitted for brevities
<PB N="28" REF="19"> &s;ake. Onely add this. That if the planes De|clination from South be above 90 <HI>gr.</HI> you mu&s;t &s;ubduct it out of 180, and the remain|der is the Declination from the North.&mdash;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 ju&s;t 90, it is then a full Ea&s;t or We&s;t plane.</P>
</DIV2>
<DIV2 N="17" TYPE="part">
<HEAD>XVII. <HI>How to draw any upright declining Diall.</HI></HEAD>
<P>FIr&s;t, draw a perpendicular or Plumb-line A B, and cro&s;&s;e it at right angles with the Horizontall line B C: and make B A equall to A &abus; in your Quadrant.</P>
<P>2 Upon the equall limbe of your Quadrant, count the planes declination (from North or South) from G, and there keep the threed: which will cut &s;ome of tho&s;e lines that are drawn within the upper &s;quare.</P>
<P>3 Ob&s;erve fir&s;t, tho&s;e inter&s;ections which the prickt lines make with the threed at <HI>b, d, m.&mdash;</HI>Take then the length from A the center of the quadrant to <HI>b;</HI> and &s;et it here upon the
<PB N="29" REF="19"> Horizontall line from B to 1, (always on that &s;ide of B, which looks to the &s;ame coa&s;t whereunto the plane declineth.) So, take from the quadrants center A, to the &s;econd prickt lines inter&s;ection with the threed, at <HI>d;</HI> and &s;et it here from B to 2. So likewi&s;e the third A <HI>m;</HI> mu&s;t be &s;et from B to 3.</P>
<P>4 Ob&s;erve again all &s;uch inter&s;ections as are made with the threed, by the re&s;t of tho&s;e lines who&s;e common concurrence is in the point M, namely, at <HI>a, c, e, h:</HI> and take their &s;everall lengths from the quadrants center A, and prick them here down on the other &s;ide B (contrary to the coa&s;t of declination) namely, at 11, 10, 9, 8. Then for the next line upon the quadrant (which doth not, but would inter&s;ect the threed, if it were drawn out far enough) ob&s;erve where the threed cuts the extravagant line <HI>r s,</HI> namely, in <HI>s:</HI> and take from A to <HI>s,</HI> and turn that length twice from B, &s;o &s;hall it de&s;igne the point 7. After|wards at the point 7, draw the infinite line CD parallel to B A. Al&s;o &s;et off the houre of 6, on that &s;ide B which is contrary to the coa&s;t of Declination, namely, from B to E, according as the angle between 12 and 6 &s;hall be found by the fifteenth.</P>
<P>5 Draw all the houre-lines from A, the
<PB REF="20" MS="y">
<FIGURE>
<HEAD><HI>Declination</HI> 28^d^. <HI>S. Ea&s;t</HI></HEAD>
<FIGDESC>diagram</FIGDESC>
</FIGURE>
<PB N="31" REF="20"> center of your Diall, through the points 3, 2, 1, 12, 11, 10, 9, 8, 7, in &s;uch wi&s;e, that as ma|ny as well can, may cut the line D C, as is here done, in <HI>p</HI> and <HI>q.</HI></P>
<P>6 Make 6, 5, equall to 6, 7: and 6, 4, equall to 6, <HI>p:</HI> and 6, 3, equall to 6, <HI>q:</HI> and draw the re&s;t of the houres, <HI>A</HI> 5, <HI>A</HI> 4, <HI>A</HI> 3. Thus you may get 12 houres, and if you extend them beyond the center, you &s;hall have the whole 24. Out of which you may make choice of &s;uch as will &s;erve your u&s;e.</P>
<DIV3 TYPE="section">
<HEAD>&para; <HI>For placing the &s;tyle,</HI></HEAD>
<P>Seek the Elevation and Deflexion by the fifth and &s;ixth. And make B F equall to the Deflexion, &s;etting the &s;ub&s;tylar line F A al|ways on that &s;ide 12 which is contrary to the coa&s;t of the planes declination. Make al&s;o F G equall to the Elevation: So F A G will be the pattern of the &s;tyle.</P>
<P>Or the threed lying &s;till at the planes de|clination upon the Quadrant as it did, Take the lea&s;t di&s;tance from the point X to the threed; and &s;et that length from B to H, and draw A H for the Sub&s;tylar. Then ma|king A H K a right angle, take the lea&s;t di|&s;tance from M to the threed, and make H K equall to this di&s;tance: So is K A H the pat|tern of your &s;tyle.</P>
</DIV3>
<DIV3 TYPE="section">
<PB N="32" REF="21">
<HEAD>&para; <HI>In all Dials,</HI></HEAD>
<P>The &s;tyle mu&s;t &s;tand ju&s;t over the Sub&s;tylar, Elevated &s;o much above it, as the Elevation (before found) commeth to.</P>
<P>In South Upright decliners, the center of the Diall is above (as in the former figure) and the &s;tyle points downward. But in North de|cliners, the center mu&s;t be below, and the &s;tyle mu&s;t point upward.</P>
</DIV3>
</DIV2>
<DIV2 N="18" TYPE="part">
<HEAD>XVIII. <HI>Of the upright full South-Diall.</HI></HEAD>
<P>THe Declination of the full South Diall is nothing. Whence it is, that</P>
<P>The angle betweene 12 and 6 is 90 degrees.</P>
<P>The line of 12 is the &s;ub&s;tylar.</P>
<P>The &s;tyles Elevation is the comple|ment of your latitude.</P>
<P>The way of pricking down the houres is (in a manner) the &s;ame with that before for decliners. No more needs <GAP DESC="illegible" RESP="KEYERS" EXTENT="2 letters"> be &s;aid of it.</P>
<P>The Erect full North plane is the &s;ame with this South. Onely the &s;tyle of this, points upwards toward the North pole, as the former doth downward towards the South pole.</P>
</DIV2>
<DIV2 N="19" TYPE="part">
<PB N="33" REF="21">
<HEAD>XIX. <HI>Of &V;pright far declining plaines.</HI></HEAD>
<P>THe&s;e Dialls are more difficult than tho&s;e other decliners mentioned in the &s;even|teenth, becau&s;e here the houres have no cen|ter or point of meeting upon the plaine. It will not be ami&s;&s;e therefore to &s;et down the whole worke in all parts of it.</P>
<P N="1">1 Draw a perpendicular or plumb-line A B, and cro&s;&s;e it at right angles with the Horizontall line B C. And make B A equall to A &Sun; in your quadrant, &s;etting A above B if the plaine decline from the South; or be|low B if it decline from North.</P>
<P N="2">2. Count the plaines Declination from South or North, upon the limbe of your qua|drant, from G: and there keep the threed.</P>
<P N="3">3 Among tho&s;e lines on the Qua|drant (who&s;e common concurrence is at M) ob&s;erve that inter&s;ection which is made by the 6^t^h houre from the quadrants center, with the threed. Take the length from the &s;ame center to that inter&s;ection, and prick it down here from B to C (and on that &s;ide B which looketh toward the South, if the plaine decline from South: or toward the
<PB N="34" REF="22"> North, if the plaine decline from North.) And draw out the lines C D E parallel to B A.</P>
<P N="4">4 Ob&s;erve again upon the quadrant that inter&s;ection which the &s;econd line from the the center makes with the threed, and take the length from the center of the quadrant there|unto, and prick it down towards C, namely from B to F.</P>
<P N="5">5 Take the lengths from the center of your quadrant to every houre point upon the &s;ide A C: and prick them all downe here, from C to 7 and 5, from C to 8 and 4, from C to 9, 10. And la&s;tly, take from the center of your quadrant to the point <HI>r,</HI> and turn that length twice from C: this double length will reach from C to 11, at E.</P>
<P N="6">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 &s;et it from A (towards B) to 10, the &s;ame way from A that 10 &s;tands from H.</P>
<P N="7">7 With the &s;ame length H 10, or A 10, go to your quadrant, and &s;etting one foot of it, on the &s;ide A C, in the fourth point from the Center, with the other (turned about) lay the threed at the remote&s;t di&s;tance, and keep it there.</P>
<PB REF="22" MS="y">
<P><FIGURE>
<HEAD>An upright Plain declining 82 deg. from South, Ea&s;tward.</HEAD>
<FIGDESC>diagram</FIGDESC>
</FIGURE></P>
<P N="8"><PB N="36" REF="23"> 8 From every point on the &s;ide A C of your quadrant, take the lea&s;t di&s;tances to the threed &s;o laid; &s;etting 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 lea&s;t di|&s;tance from <HI>r</HI> to the threed being twice turn|ed from A towards B, will give the length from A to 11.</P>
<P N="9">9 For the fini&s;hing then of the houres, you have no more to do, but draw right lines through each couple of corre&s;pondent 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.</P>
<DIV3 TYPE="section">
<HEAD>&para; <HI>Concerning the forming and placing of the &s;tyle.</HI></HEAD>
<P>10. BY the precedent &s;eventh propo&s;ition you may finde the plaines difference of Longitude, which (for this plaine that de|clines 82 <HI>gr.)</HI> will be (here at <HI>London)</HI> 83 <HI>gr.</HI> 43 <HI>min.</HI> and that from the South, becau&s;e the plaine declines from the South. The comple|ment of which longitude (83 <HI>gr.</HI> 43 <HI>min.)</HI> is 6 <HI>gr.</HI> 17 <HI>min.</HI> Take then fir&s;t, the length from C to 7 the next hour point upon C E, and
<PB N="37" REF="23"> carrying that extent to your quadrant, &s;et one foot of it upon 15 in the &s;cale A P: and lay the threed &s;o, that the other foot turned about may ju&s;t touch or pa&s;&s;e over it: and keep the threed there. Then (in the &s;cale A P) count the forementioned complement, 6 <HI>gr.</HI> 17 <HI>min.</HI> and taking the lea&s;t di&s;tance from that point to the threed, &s;et it from 6 a clock at C, to|wards E if the plaine decline from South, (or towards D if the plane decline from North) as you &s;ee it done here, at G. Secondly, do the &s;ame worke again upon the line A B. That is; Take from A to 7 the neere&s;t houre point, and &s;et one foot of that extent upon 15 in the &s;cale A P, and with the other foot turned a|bout, lay the threed as before. Then in the &s;ame &s;cale A P, count the &s;ame number 6 <HI>gr.</HI> 17 <HI>min.</HI> and taking the lea&s;t di&s;tance from thence to the threed, &s;et that length from A to K, an&s;wering to C G. And la&s;t of all, draw the right line G K. This &s;hall be the line of deflexion, over which the &s;tyle mu&s;t &s;tand.</P>
<P>11 Furthermore. Through the points G and K (or any other two points of the &s;ame line) draw the two lines G O, K P, both per|pendicular to the deflexion line G K. Then con&s;idering that every houre comprehends 15 degrees of longitude (that is, that from C to 7
<PB N="38" REF="24"> is 15, and from 7 to 8 is 15, &amp;c.) and &s;ince that C G is 6 <HI>gr.</HI> 17 <HI>min.</HI> If <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter"> G, be taken out of C 7 which is 15 <HI>gr.</HI> there will remain G 7, 8 <HI>gr.</HI> 43 <HI>min.</HI> To which, if you add from 7 to 9, which is two houres or 30 degrees, the &s;um will be 38 <HI>gr.</HI> 43 <HI>min.</HI> who&s;e complement is 51 <HI>gr.</HI> 17 <HI>min.</HI> If now you make the angles G M R, and K N S, each 51 <HI>gr.</HI> 17 <HI>min.</HI> they will cut the Deflexion line G K, in R and S. And if further, to the radius G R, you de&s;cribe the arke R T; and to the radius K S you de|&s;cribe the arke S V; and draw the line T V, a tangent to both the&s;e arkes, the <HI>Trapezium</HI> G T K V &s;hall be the pattern of your &s;tyle. In placing which, you mu&s;t be carefull that the&s;e perpendicular lengths G T and K V (perpendicular I &s;ay to T V the Fi|duciall edge) be ju&s;tly placed upon the two a&s;&s;umed points at G and K.&mdash;Or having found G 7 to be 8 <HI>gr.</HI> 43 <HI>min.</HI> you may add to it from 7 to 10, which is (three houres or) 45 degrees. The &s;umme will be 53 <HI>gr.</HI> 43 <HI>min.</HI> who&s;e complement is 36 <HI>gr.</HI> 17 <HI>minutes.</HI> If now from the points O and P (where the &s;aid houre of 10 cuts the two f<GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">re-mentioned per|pendiculars G O and K P) you make the an|gles G O R and K P S, each equall to 36 <HI>gr.</HI> 17 <HI>min.</HI> they will cut the deflexion line G K in
<PB N="39" REF="24"> the &s;ame two points R and S. After which, you may proceed to make the patterne of your &s;tyle, as before.</P>
<P>&para; 1 Note that in performing the fifth &s;ection of this propo&s;ition, in&s;tead of taking tho&s;e houre points from the Center of your quadrant upon A C the &s;ide of your quadrant (if tho&s;e di&s;tances &s;hould be too great for your plaine) you may lay the threed any where upon the Quadrant, and in|&s;tead of taking from the center to the fore-named points, you may take the lea&s;t di&s;tances from the &s;aid points to the threed, &s;everally, and &s;et them down from C to 7 and 5, and from C to 8 and 4, and &s;o to 9, 10; and for 11, you mu&s;t take from the point <HI>r</HI> to the threed, and &s;et it twice from C; by which meanes they will all be of le&s;&s;e di&s;tance from C. And then all the worke is to be con|tinued, as is before pre&s;cribed.&mdash;Or if the &s;aid di&s;tances &s;hould be too little, you may double, triple, or, &amp;c, to make them greater.</P>
<P>&para; 2 Note again, that in decliners from the North, that difference of Lon|gitude
<PB REF="25"> which you finde by the &s;e|venth, is to be reckoned from the North, and &s;o the complement of it is to be accounted from C (or 6 a clock) towards D. And that the wide&s;t parts of the houres in the&s;e North plaines mu&s;t point upwards, and the clo&s;e&s;t parts downwards; contrary to what is expre&s;&s;ed here in this plane, which hath its Declination from the South.</P>
<P>&para; 3 Note la&s;tly, that this direction here given for enlarging the houres in farre Decliners, may ea&s;ily be ap|plied to &s;uch direct, or Horizontall Dials (as are mentioned in the 26 follwoing) upon which the pole hath but &s;mall Elevation. For the Diall (or onely &s;ome chief houres of it) being de&s;cribed in its naturall &s;treightne&s;&s;e, may be enlarged by the &s;ame meanes that this la&s;t was. Which will not be hard to do, but would be tedious here to run over againe.</P>
</DIV3>
</DIV2>
<DIV2 N="20" TYPE="part">
<PB N="41" REF="25">
<HEAD>XX. <HI>Of full Ea&s;t and We&s;t upright Dialls.</HI></HEAD>
<P>THe&s;e are more ea&s;ie than the former &s;ort were. For having drawn the plumb-line A B, and a&s;&s;umed the point A for the houre of 6; goe to your Quadrant, and take from the center of it to all the houre-points upon the &s;ide A C; and prick the fir&s;t of them down in the line A B, from A to 5 and 7: the &s;econd from A to 4 and 8; the third from A to 3: the 4^t^h. from A to 2. And for the fifth, Take from the center of your Quadrant to the point, <HI>r,</HI> and &s;et that length twice from A, &s;o it &s;hall limit out the point 1.&mdash;Having the&s;e points, draw lines through them, all pa|rallel one to the other, and all pointing up to the North; namely &s;o, as to make the acute angles B A C equall to the complement of your latitude.</P>
<PB REF="26" MS="y">
<P><FIGURE>
<HEAD>A full We&s;t upright diall.</HEAD>
<FIGDESC>diagram</FIGDESC>
</FIGURE></P>
<DIV3 TYPE="section">
<PB N="43" REF="26">
<HEAD>&para; <HI>For the Style.</HI></HEAD>
<P>IT mu&s;t always &s;tand over the line of 6 a|clock, parallel to it, and di&s;tant every where from it according to the length of A D. Which lenght is &s;oone found, by drawing A D perpendicular to the houre-lines, cut|ting the third houre from 6, in D. By which line you may make the patterne of your &s;tyle. For the Fiduciall edge lyes parallel to the line of 6, A C, and at the di&s;tance of that line A D.</P>
<P>&para; 1 Note here too; that if your lengths from the Quadrants center to the houre-points be too long, you may &s;horten them by laying the threed upon the Quadrant according as your convenience &s;hall direct, and taking the lea&s;t di&s;tances from tho&s;e houre-points to the threed; and &s;o pricking them on from A or 6, to 5, 4, 3, &amp;c; as was before mentioned in the fir&s;t Note upon the former propo&s;ition.&mdash;Or if they be too little, they may be doubled, &amp;c. as is there expre&s;&s;ed.</P>
<PB N="44" REF="27">
<P> &para; 2 Note further, that what is here done for de&s;cribing the&s;e Ea&s;t and We&s;t Dialls, may be applied to the direct Polar Plaine. Onely re|member that you are not tyed (in the Polar) to make the houres to any &s;et angle with the line B A, but they are be&s;t at right angles; for then the line A B may be taken for, and placed as, the Horizontall line of the &s;aid plaine; all the houres lying as verticall lines unto it. And al&s;o the line of 6 here, mu&s;t be ta|ken (in the direct polar) for the line of 12, and the re&s;t of the houres are to be drawn alike on both &s;ides 12: nothing in &s;ub&s;tance differing from the&s;e Ea&s;t and We&s;t Planes.</P>
</DIV3>
</DIV2>
<DIV2 N="21" TYPE="part">
<HEAD>XXI. <HI>In Ea&s;t and We&s;t re-in-cliners, To get the Deflexion.</HI></HEAD>
<P>COunt the re-in-clination from D to|wards C. Take the lea&s;t di&s;tance from thence to the &s;ide A C. Set that lenght
<PB N="45" REF="27"> <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">rom M to Y, and lay the threed at Y. The <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">egrees F K will give the Deflexion.</P>
<P>The &s;ub&s;tylar line mu&s;t a&s;cend in Recliners, <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">nd de&s;cend in Incliners, from the line of 12, <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">ccording to the quantity of this Deflex|<GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">on.</P>
<P>The line of 12 lyes always Parallel to the Horizon.</P>
</DIV2>
<DIV2 N="22" TYPE="part">
<HEAD>XXII. <HI>To finde the angle Between</HI> 12 <HI>and</HI> 6.</HEAD>
<P>COunt the Re-in-clination From E to|wards B. The threed there laid will cut the equall limbe. The degrees whereof from G to the threed, are the angle required.</P>
</DIV2>
<DIV2 N="23" TYPE="part">
<HEAD>XXIII. <HI>To get the Styles Elevation.</HI></HEAD>
<P>LAy the threed to the Re-in-clination numbered in the equall limbe from F, and take the lea&s;t di&s;tance from N to the threed. Set one foot of that length in B, and <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">ay the threed &s;o as to touch the other foot when it is turned about. The threed &s;o laid, gives the Elevation in the equall limbe, <GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter">rom F.</P>
</DIV2>
<DIV2 N="24" TYPE="part">
<PB N="46" REF="28">
<HEAD>XXIV. <HI>To finde the difference of Longitude.</HI></HEAD>
<P N="1">1 COunt the Deflexion in the equall limbe from F, and lay the threed to it; and take the lea&s;t di&s;tance from B to the threed. Put one foot of this length in N, and apply the threed to the remote&s;t di&s;tance of the other foot. The threed will then &s;hew in the equall limbe, the difference of longitude, if you count from F.</P>
<P N="2">2 Count the deflexion in the equall limbe, from G: and to the threed there laid, take the lea&s;t di&s;tance from B. Mea&s;ure that length upon the &s;ide A B, from A; keeping one foot there fixed. Then lay the threed to the plaines Re-in-clination counted al&s;o from F in the equall limbe, and take the lea&s;t di&s;tance from your &s;tanding to the threed. Set one foot of this length in B, applying the threed to the other foot turned about. The threed &s;o laid, gives the difference of Longi|tude in the equall limbe, from G.</P>
<P>Thus if an Ea&s;t or We&s;t plaine Re-incline, here at <HI>London,</HI> 30 degrees, it will have in
<PB N="47" REF="28">
<TABLE>
<ROW>
<CELL>Deflexion</CELL>
<CELL>47&deg;. 26&prime;.</CELL>
</ROW>
<ROW>
<CELL>Angle from 12 to 6.</CELL>
<CELL>55. 26.</CELL>
</ROW>
<ROW>
<CELL>Elevation</CELL>
<CELL>23&deg;. 02&prime;.</CELL>
</ROW>
<ROW>
<CELL>Differ. of long.</CELL>
<CELL>70. 14.</CELL>
</ROW>
</TABLE></P>
</DIV2>
<DIV2 N="25" TYPE="part">
<HEAD>XXV. <HI>How to draw the Diall.</HI></HEAD>
<P>VPon the back&s;ide of your Quadrant, in the upper part of it, you have lines drawne altogether like tho&s;e on the fore&s;aid placed neer the Quadrants center, the u&s;e of which was &s;hewed before.</P>
<P>The manner of work in this propo&s;ition is in mo&s;t things &s;utable to that in the &s;even|teenth, and will need no other direction.</P>
<P>Onely for placing the lines, Take notice, that,</P>
<P>The line of 12 in the&s;e Ea&s;t and We&s;t Re|in-cliners, lyeth alwayes parallel to the Ho|rizontall line of the plaine. So that if we &s;uppo&s;e the former figure of the &s;eventeenth to repre&s;ent one of the&s;e Dialls, then A B mu&s;t be conceived to lie Horizontall, and B C verticall. All other works will be like to tho&s;e in the &s;eventeenth.</P>
<P>The &s;tyle in recliners pointeth upward, and the &s;ub&s;tylar and houre of 6 do a&s;cend above
<PB N="48" REF="29"> the line of 12, So much as the Deflexion and angle from 12 to 6, come to. The center of the Diall is on the South end of the line of 12.</P>
<P>The &s;tyle in incliners pointeth downward, and the &s;ub&s;tylar and houre of 6 do de&s;cend below the line of 12, &s;o much as the Deflex|ion and angle from 12 to 6 come unto. The center of the Diall is on the North end of the 12 a clock line.</P>
<P>The&s;e things being ob&s;erved, you mu&s;t count the Re-in-clination of your plaine in the equall limbe on the back&s;ide from the left hand toward the right, according as the figures are &s;et: and there lay the threed and keep it. Then ob&s;erve how it cuts the lines next to the center, and proceed in all things as in the &s;eventeenth before.</P>
<P>&para; Note that you may finde the inclination of a plaine by applying one &s;ide of your Qua|drant to the plaines verticall line: for &s;o the threed will cut the quantity of inclination in the degrees of the equall limbe being num|bered from that &s;ide of the Quadrant which toucheth the plaine.&mdash;And for find|ing the reclination, you may lay a ruler to the verticall line of the reclining face, and take the inclination of the under &s;ide of that ruler.
<PB N="49" REF="29"> That inclination will be the &s;ame with the re|clination.</P>
<P>Note al&s;o, that this here delivered for Ea&s;t and We&s;t Re-in-cliners, is intended chiefly for drawing houres upon tho&s;e kindes of plaines when you meet with them upon Bo|dies cut regularly. For otherwi&s;e you will hardly ever finde any &s;uch ju&s;t plaine upon a fixed building.</P>
<P>La&s;tly, for a Scale of chords, which here, and in &s;ome of the precedent precepts is re|quired, you may make u&s;e of the equall limbe of your quadrant.</P>
</DIV2>
<DIV2 N="26" TYPE="part">
<HEAD>XXVI. <HI>To make an Horizontall Diall to any Latitude.</HI></HEAD>
<P>FIr&s;t, draw the right line B C, and erect the Perpendicular A H. Then take from the center (on either &s;ide of your quadrant) to the third houre upon the &s;ide A C; and make A H equall thereto. And draw F H parallel to B C; and the line 5 K 7 al&s;o ju&s;t in the mid&s;t of them.&mdash;After this, lay the threed to the Latitude of the plaine counted in the equall limbe: and take from every point of th<GAP DESC="illegible" RESP="KEYERS" EXTENT="1 letter"> &s;ide A C, the lea&s;t di&s;tance to the
<PB REF="30">
<FIGURE>
<FIGDESC>diagram</FIGDESC>
</FIGURE>
<PB N="51" REF="30"> threed, and &s;et each of them down both wayes, 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 <HI>r</HI> upon the &s;ide A C, to the threed, and &s;et that length from K to 5 and 7, both wayes.&mdash;You have now nothing more to do, but onely from H to draw the hour-lines to all the fore|named points: &s;o the draught is ea&s;ily fini&s;hed.</P>
<P>The &s;tyle mu&s;t &s;tand upon the line of 12, and is to be elevated according to the plaines latitude: as the manner is in all Horizontall Dialls.</P>
<P>&para; The u&s;e of this propo&s;ition is to draw all Dialls in any Latitude for any direct re-in-clining plaine. For, the re-in-clination com|pared (in North re-in-cliners) with the poles Elevation: or (in South direct re-in-cliners) with the Equinoctialls Altitude, will ea&s;i|ly give the plaines Latitude: in the former, the difference was the Elevation it &s;elfe: in the latter, the Complement of the poles Ele|vation.&mdash;And this propo&s;ition, with the &s;eventeenth for upright plaines; the twen|tieth for upright Ea&s;t and We&s;t, and &s;o al&s;o for Polar plaines on which the pole hath no Elevation: the twenty fifth for Ea&s;t and
<PB N="52" REF="31"> We&s;t re-in-cliners: the eighteenth for full North and South erect; will furni&s;h you with wayes to draw Dialls upon &s;uch regu|lar bodies, who&s;e plaines have any &s;uch of the fore-mentioned A&s;pects.</P>
</DIV2>
<DIV2 N="27" TYPE="part">
<HEAD>XXVII. <HI>To finde the houre of the Night by the Starres.</HI></HEAD>
<P>THe Stars upon the Quadrant (one or other of them) will alwayes be in convenient place of the heavens: that is, of two or more houres di|&s;tance from the Meridian.&mdash;Having then made choice of that Starre that is fitte&s;t, looke what number is annexed to the name of it. Seeke that number in the left margin of the fore&s;ide of your quadrant, and clo&s;e by the houre lines, and rectifie the Bead to it.&mdash;Then hold up the quadrant &s;teadily, with the &s;ights levelled to the Starre, as if you were to take the Starres Altitude: and you &s;hall finde the Bead to &s;hew (among the Summer houres of the quadrant) the motion of the Starre in houres, quarters, and parts of
<PB N="53" REF="31"> a quarter. This is called the Starrs houre; but this is not the houre of the Night till it be turned into the Suns houre: which thing is to be done in this manner.</P>
<P>Look upon the back-&s;ide of the quadrant for your Starre, and lay the threed upon it: &s;lipping the Bead down to the &s;lope houres below, till it &s;tand upon the &s;ame quarter and part (from &s;ome ju&s;t houre on the left hand of the bead) with the Starres houre before found. Then note the &s;aid houre on the left hand which goeth next before the bead, for that mu&s;t be &s;uppo&s;ed to repre&s;ent the Starres houre, and mu&s;t therefore be called by the &s;ame, or number that the Starres houre was. And following houres (from the Bead towards the right hand, mu&s;t &s;ucce&s;&s;ive|ly take their numbers, untill you come to be under the day of your moneth. Unto which day if the threed be layd, the bead will (by keeping your former account) &s;hew the true houre, quarter, and part, of the Night.</P>
<P><HI>Example,</HI> 1. On <HI>January</HI> the 20, the hour of <HI>Cor Leonis</HI> was ob&s;erved Ea&s;tward of the Me|ridian, to be 9 and &frac14; and &frac13; part a quarter. The threed laid upon that Star, on the back-&s;ide of the Quadrant, will cro&s;&s;e the &s;lope houres as doth the line A B. And the bead put
<PB N="54" REF="32"> down to the forementioned parts of the houre, will &s;tand at the point B. So that the hour C mu&s;t be called 9 a clock, which is the ob|&s;erved houre of the Starre. Then the line D mu&s;t be called 10 a clock: and the threed being put to <HI>January</HI> 20 (taken in the lower circular line of moneths) will lie in the line A E; and the Bead at E &s;hews the time of the Night to be pa&s;t (the line D, that is, pa&s;t) 10 a clock, about &frac14; and &frac13; part of a quarter, which is 15 and 5 <HI>minutes:</HI> or 20 <HI>minutes</HI> pa&s;t 10 at night.&mdash;But if this ob|&s;ervation had been upon the &s;econd day of <HI>November:</HI> then the threed layd upon (the day given in the lower circle of moneths) <HI>November</HI> 2, would lie in the line A F: and the Bead would be upon the full houre|line that pa&s;&s;eth 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 &s;o forward till your account fall up|on F: which mu&s;t be 4 a clock pa&s;t (12 or) Midnight.</P>
<P><HI>Example</HI> 2. Upon the 8 of <HI>Augu&s;t,</HI> the Starre <HI>Aquila</HI> was &s;een on the We&s;t&s;i de of the Meridian, and the houre of it was found, 3 and &frac12; an houre and &frac12; a quarter. The threed therefore being laid upon that Starre
<PB N="35" REF="32"> would be as the line A G, and the Bead (re|ctifyed to the &frac12; houre and &frac12; quarter) would &s;tand at the point G. So that the next houre|line on the left hand of G, mu&s;t be called 3 a clock: and the line F mu&s;t be 8 a clock. Then, the threed being removed to the day of your moneth <HI>(Augu&s;t</HI> 8, in the upper circular line of moneths) will lye in the line A B: and the Bead at B will &s;hew the houre of the Night (if you keep your former account) to be &frac14; and halfe pa&s;t 1 a clock. For if F be 8 a clock (as is before expre&s;&s;ed) then the la&s;t houre of the limbe is 11, the fir&s;t is 12, the &s;econd 1; beyond which, the Bead B is about 22 <HI>minutes</HI> of an houre. There|fore the houre of the night is 1 a clock 22 <HI>mi|nutes.</HI></P>
<P>By the&s;e examples the manner of the worke will &s;ufficiently appear in all ca&s;es.</P>
</DIV2>
<DIV2 TYPE="part">
<PB N="56" REF="33">
<HEAD>The v&s;e of the Altimetrie Scale.</HEAD>
<P>THat Scale on the fore-&s;ide of the Quadrant next to the equall limbe is here called the Altimerie Scale. It is numbred by 1, 2, 3, &amp;c, to 10, 20, 30, &amp;c, to 100. Each of which num|bers are be&s;t &s;uppo&s;ed to be 100 fold. <HI>viz,</HI> 100, 200, &amp;c, to 1000, 2000, &amp;c: to 10000: and all the le&s;&s;er parts e&s;timated accordingly.&mdash;The ground on which you &s;tand to make your men&s;uration, is al&s;o &s;uppo&s;ed to be a ju&s;t levell.</P>
<DIV3 N="1" TYPE="section">
<HEAD>I. <HI>To finde any height at one ob&s;ervation.</HI></HEAD>
<P>LEt your &s;tation be at E; and the &s;ights D A directed to the point F: the threed A B cutts off the parts C B in the mea&s;uring Scale: which parts mu&s;t be remembred.&mdash;Then mea&s;ure from your &s;tation E, to the point H, which is ju&s;t under F. And (al|wayes in this ca&s;e (multiply this di&s;tance
<PB N="57" REF="33"> 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 mu&s;t add H G, or D E, the height from your eye at D to your foot at E.</P>
<P>Thus if the threed A B &s;hould cut off C B 1500 parts, and the di&s;tance E were 59 feet, The height G F would be 88. 500, or 88&frac12; feet.</P>
</DIV3>
<DIV3 N="2" TYPE="section">
<HEAD>II. <HI>To finde part of an Altitude.</HI></HEAD>
<P>LEt the length of F X be onely required. Standing then at E, you may finde the altitude G F. Keep &s;till the &s;ame &s;tanding at E, and finde the altitude G X, by the la&s;t precedent. So G F taken from G X, gives F X required.</P>
</DIV3>
<DIV3 N="3" TYPE="section">
<HEAD>III. <HI>Standing upon a known height, to finde a Di&s;tance.</HI></HEAD>
<P>LEt the height F H be known, and the di|&s;tance H K be required. Order your &s;tand|ing &s;o, that the two &s;ights P, S; the point F, and the di&s;tance K, may all appeare in one
<PB REF="34">
<FIGURE>
<FIGDESC>diagram</FIGDESC>
</FIGURE>
<PB N="59" REF="34"> right line. Then look what degrees the plum|met cuts off in the equall limbe from Q. Count the &s;ame number in the &s;ame limbe, from S; and there lay the threed, as P T. Note then, what parts it cuts upon the mea&s;uring Scale from Q to T. Multiply tho&s;e parts in|to F H the known Altitude: and from the product cut off three figures, the remainder or quotient, is the di&s;tance H K.</P>
<P>Thus, if the threed P R &s;hould cut off Q R in the equall limbe 56&frac13; degrees, the &s;ame counted the other way, from S to T in the equall limbe, and the threed laid there|to, would give 667 in the mea&s;uring Scale. Then F G being 88&frac12; feet, and G H (&s;uppo&s;e) five feet, F H mu&s;t be 93&frac12; feet. This mul|tiplyed into 667, makes 62364: from whence cutting away the three right hand figures, there remaines 62. 364 or 62&frac13; feet, for the di|&s;tance H K.</P>
</DIV3>
<DIV3 N="4" TYPE="section">
<HEAD>IV. <HI>To finde part of a di&s;tance.</HI></HEAD>
<P>IF the di&s;tance of K from Z were requi|red. Fir&s;t, finde H K, then H Z, by the third precedent: their difference is K Z. If
<PB N="80" REF="35"> K Z were a trench, you might from the tower F, finde the breadth of it without any approach unto it.</P>
</DIV3>
<DIV3 N="5" TYPE="section">
<HEAD>V. <HI>To finde a height at two ob&s;ervations.</HI></HEAD>
<P>IF F H were to be mea&s;ured, and the way from E to H were unpa&s;&s;eable, &s;o that the di&s;tance of E from H could not be mea&s;ured. You mu&s;t in this ca&s;e make two ob&s;ervations. For which purpo&s;e, Take your fir&s;t &s;tation at E, and direct the &s;ights D, A, to the point F: noting what parts the threed cuts upon the equall limbe from C to B. Then goe back|wards in a right line, to a competent di&s;tance, as to M; and there making a &s;econd &s;tation, ob&s;erve (as before) what degrees the threed cuts upon the equall limbe from N to O: (the two &s;ights L, I, being ju&s;tly directed to the point F). Then count the&s;e two arkes in the equall limbe, from the contrary &s;ide of the quadrant, namely, from D to Y, and from L to <HI>u,</HI> and applying the threed thereto, looke what parts it cuts from the mea&s;uring Scale at Y, and V. Take the le&s;&s;er number of parts, out of the greater, noting the Difference. Mea&s;ure al&s;o the di&s;tance of
<PB REF="35"> your two &s;tations, namely from E to M, and add three cyphers to that mea&s;ure. This la&s;t number mu&s;t (in this kinde of worke) be di|vided alwayes, by the fore-noted difference: and the quotient will give the Altitude of F above G.</P>
<P><HI>Example.</HI> Let the fir&s;t ob&s;ervation cut off 38&frac23; <HI>gr.</HI> in the equall limbe. The &s;econd 56&frac13; <HI>gr.</HI> Count the fir&s;t arke from D to Y: the threed there laid gives 1250 in the mea|&s;uring Scale. The &s;econd &s;o counted from L to V, gives 667. The difference of the&s;e two, is 583. Let the di&s;tance of the &s;tations mea|&s;ured from E to M, be 5160 feet. This num|ber, with three cyphers added, is 5160000. Which divided by 583 (the former difference) gives in the quotient, 8850 or 88&frac12; feet for the height G F. And if G H be 5 foot more; The whole height H F will be 93&frac12; feet.</P>
<P>&para; Note that in the&s;e men&s;urations, the point G is &s;uppo&s;ed to &s;tand in the &s;ame levell with the corner of your Qua|drant D and L. So that G H, D E, L M, are all of one height. And note too, that the two &s;tationary points are E and M, namely tho&s;e which are ju&s;t under the corners D and L.</P>
</DIV3>
</DIV2>
<TRAILER>FINIS.</TRAILER>
</DIV1>
</BODY>
<BACK>
<DIV1 TYPE="errata">
<PB REF="36">
<HEAD>Errata.</HEAD>
<P><TABLE>
<ROW>
<CELL><HI>Pag.</HI></CELL>
<CELL><HI>Line.</HI></CELL>
<CELL><HI>Read.</HI></CELL>
</ROW>
<ROW>
<CELL>6</CELL>
<CELL>1</CELL>
<CELL>Ninetyeth.</CELL>
</ROW>
<ROW>
<CELL>11</CELL>
<CELL>21</CELL>
<CELL>the threed laid at K.</CELL>
</ROW>
<ROW>
<CELL>14</CELL>
<CELL>25</CELL>
<CELL>horizon, is 29 <HI>gr.</HI> 21 <HI>min.</HI></CELL>
</ROW>
<ROW>
<CELL>16</CELL>
<CELL>19&middot;20</CELL>
<CELL>cuts from P. Count</CELL>
</ROW>
<ROW>
<CELL>25</CELL>
<CELL>13</CELL>
<CELL>Count the plaines</CELL>
</ROW>
<ROW>
<CELL>26</CELL>
<CELL>19</CELL>
<CELL>the Suns altitude.</CELL>
</ROW>
<ROW>
<CELL>34</CELL>
<CELL>2</CELL>
<CELL>the line C D E</CELL>
</ROW>
<ROW>
<CELL>52</CELL>
<CELL>16</CELL>
<CELL>quadrant, clo&s;e by</CELL>
</ROW>
<ROW>
<CELL>53</CELL>
<CELL>15</CELL>
<CELL>the &s;ame name or number</CELL>
</ROW>
<ROW>
<CELL>53</CELL>
<CELL>25</CELL>
<CELL>&frac13; part of a quarter.</CELL>
</ROW>
<ROW>
<CELL>56</CELL>
<CELL>1</CELL>
<CELL>Altimetric</CELL>
</ROW>
<ROW>
<CELL>56</CELL>
<CELL>4</CELL>
<CELL>Altimetric.</CELL>
</ROW>
</TABLE></P>
<PB REF="36">
</DIV1>
</BACK>
</TEXT>
</EEBO>
</ETS>
