Posthuma Fosteri: THE DESCRIPTION OF A RULER, Upon which is inscribed divers SCALES: AND The uses thereof: Invented and written by Mr. SAMVEL FOSTER, Late Professor of astronomy in gresham-college. By which the most usual Propositions in Astronomy, Navigation, and Dialling, are facilely performed. Also, a further use of the said Scales in delineating of far declining Dials; and of those that Decline and Recline three several ways. With the delineating of all horizontal Dials, between 30 and 60 gr. of Latitude, without drawing any lines but the hours themselves. LONDON: Printed by ROBERT & WILLIAM LEYBOURN, for NICHOLAS BOURN, at the South entrance into the royal Exchange, 1654. TO THE READER. COURTEOUS READER, WE here present to thy view, this short TREATISE, (written by that learned Professor of astronomy in Gresham college, Mr. SAMUEL FOSTER deceased) containing in it the Description and use of certain Lines to be put upon a straight Ruler, in the ready solution of many necessary Questions, as well geometrical, as belonging to astronomy, Navigation, and Dialling. We should not thus hastily have thrust this into the World without its fellows, had we not been assuredly informed that some people, greedy rather of unjust gain to themselves, then with honesty to sit still, had prepared one for the press, from a spurious and imperfect copy, both to the abuse of thee, and discredit of the industrious Author: who had he thought such things as these worthy him or the press, could have daily crammed thee with them, to his own loss of time, and thy satiety. However, such as it now is, we assure thee was his own, and doubt not, but thou wilt find it pleasant in the use, profitable to thee, and portable in it self. We thought fit farther to advertise thee, that there are abroad in particular hands, imperfect Copies of some other Treatises of the same Author: Namely, An easy Geometrical way of Dialling. Another most easy way to project hourlines upon all kind of Superficies, without respect had to their standing, either in respect of Declination or Inclination. A Quadrant fitted with lines for the solution of most Questions of the Sphere: with some other things of the like nature. We fear least sinister ends of some mean Artists, or ignorant mechanics, (for those of ingenuity in whose hands they may be, we no way distrust) may engage them to father these things as their own, or at least under the author's name put out lame and imperfect Copies of otherwise good things: To prevent which we give thee this timely notice, assuring thee, that these, together with divers other pieces never yet seen, except by very few, and if we deceive not ourselves, of much greater weight, are making ready for the press by the author's approbation, and from his own copy in our command, with his other papers, of which thou shalt be made partaker within few months. In the mean time, we desire thee not to lose thy time in reading, or money in buying any the forementioned Treatises put out by any other, either under their own, or our author's name, except such as shall be attested by me, who am one of those entrusted for that purpose, and who shall be ever studious of thy good. Gray's inn, July 26, 1652. Edm. Wingate. The description of the Scales on the Ruler. THere are 9 Scales upon the Ruler. 1. Of Inches or equal parts. 2. Of horizontal spaces: with 3. A Scale of 60 Chords fitted to the same Radius, proper to that horizontal Scale. 4. A Scale of Sines, to a Radius of two Inches. 5. A Scale of Secants: and 6. Of Tangents, both of the same Radius with the Sines. All these are upon one side. On the other side is, 7. A little Scale of 60 Chords, of the same length with the common Radius of Sines, Secants, Tangents, and common in use to them all. 8. A large Scale of Versed-Sines of the whole length of the Ruler, with a Zodiac annexed to it. 9 A Scale of unequal parts divided into 90, noted with ☉, of two Inches Radius as is the line of Sines. THE uses of the SCALES on the RULER. CHAP. I. Of the Scale of equal parts. THe Scale of Inches is a Scale of equal parts, and will perform (by protraction upon paper) such conclusions as are usually wrought in Lines and Numbers, as in Master gunter's 10. Prop. 2. Chap. Sector, may be seen, and in others that have written in the same kind. An Example in Numbers like his 10th. Prop. As 15 to 5, So 7 to what? But if your second term shall be greater than the first, than the form of working must be changed; as in this Example. As 5 to 15, So 20 to what? Upon the line AB, I set the second term 15, which here suppose to be AD: then with the first term 5, upon the centre D, I describe the ark GH, and draw AG that may just touch it. Again, having taken 20 the third term, out of the same Scale, I set one foot of that extent upon the line AB, removing it till it fall into such a place, as that the other foot being turned about will justly touch the line AG before drawn, and where (upon such conditions) it resteth, I make the point C. Then measuring AC upon your Scale, you shall find it to reach 60 parts, which is the fourth number required. The form of work (though not so geometrical) is here given because it is more expedite than the other by drawing parallel lines. But in some practices, the other may be used. I have been the more large upon this, because the solutions of proportions which follow must be referred hither, the form of their operations being the same with this. In them therefore shall only be intimated what must be done in general, the particular way of working being here explicated. CHAP. II. Of the Scales of Chords. THe Scales of Chords are to protract and measure angles. The manner how they must be used is well enough known. Only note here, that you may make the line of Sines, the line of Versed Sines, or the zodiac, (beginning at the middle of these two last mentioned) to serve for Chords of several extents, if you count each half degree for a whole degree, and so double all the numbers, accounting 10 to be 20; and 30 to be 60, 45 to be 90, &c. By this you are fitted with several Scales of Chords which are of different lengths, and may be used, each of them, as occasion shall require. And (by the way) the Versed Scale being taken for Chords, it will be of the same Radius or length with the Sines, Tangents and Secants and so will protract Angles to a Circle of their Radius, which is useful in Projections, and many other things: and so the little Scale of 60 Chords might be spared. CHAP. III. The joint use of equal parts and Chords. BY these two together, may be resolved all Cases in plain Triangles without proportional work, if the three quantities given be protracted by help of these two Scales. For their principal uses are to measure lines and angles. Here must be remembered. First, that if the three angles alone be given, then will the proportions only of the sides be found, but not the sides themselves. Secondly, that if two angles be known, than is the third also known; because it is the residue of the sum of the other two to 180 gr. Then with a line of Chords protract the angles at A and B according to their known quantities, so shall the two legs of the same angles meet at C: And if the length AC be taken and measured upon your Scale of equal parts, the same will show about 1004, intimating that C is from A 1004 feet. So again BC being measured in the same Scale will give 1120; showing that from B to C are about 1120 feet. In this manner may perpendicular altitudes (as of Towers or such like) be measured, though no access can be had to them; and that much better than by the geometrical Square. For it is not here requisite that the ground whereon the mensuration is made, should be level, as if you work by the Square it is most commonly required; neither are you tied to right angled Triangles here, as there you are. As for example, If one station be at A, and the other at B, you may, by the precedent work get the distance AD. Then (standing at A) observe the altitude of C, the compliment of that altitude gives the angle ACD. And again from A, if you observe the altitude of D, the difference of these two altitudes observed, gives the angle god. Or if D had appeared lower than your station, than the sum of your two observed altitudes had made the angle god. However, you have now the three angles and side AD; you may therefore, by help of them, find the length of CD. In such manner may all cases in plain Trigonometry be resolved. CHAP. IV. Of the Scales of Sines, Secants, and Tangents. THese being jointly used with a Scale of equal parts, will resolve all things in plain Triangles, by working such proportions as are usually given for that purpose. The manner of the work may be gathered by the former delineation in the 1 Chapt. For if AC and AD had been taken out of the Scale of Sines, or Tangents, or Secants; and CE, DG, out of the Scale of equal parts; then had the work been resolved in Sines and equal parts, or Tangents and equal parts, &c. And so this kind of work will produce the quantity required; although there be no delineation of the particular parts of the Triangle, as was before done by protraction. By these same three Scales of Sines, Tangents, and Secants, may be wrought all things pertaining to spherical Triangles. That is to say. 1. Proportions in Sines alone. 2. Proportions in Tangents alone. 3. Proportions in Secants alone. 4. Proportions in Sines and Tangents together. By natural Sines and Tangents. By ☉ and Versed-Sines. 5. Proportions in Sines and Secants together. 6. Proportions in Tangents and Secants together. An Example in Sines alone What Declination shall the Sun have in the 10 gr. of Aries? Upon the line AB (see Chap. 1.) set the Radius or Sine of 90 AC: and make AD equal to the Sine of 10 gr. (which is the sun's distance from the next equinoctial point.) Then with the Sine of 23½ (taken out of the same Scale of Sines) upon the centre C, describe the ark EF; to which, from A, draw the Tangent line AE. Lastly, from D, to this line AE, take the least distance, the same measured in the line of Sines, gives about 4 gr. for the declination required. The proportion that is here wrought stands thus. As the Radius, to the Sine of 23 ½; So the Sine of 10, to the Sine of 4 gr. The like manner of work is to be used in Sines and Tangents (or any of the other two) joined together; if it be remembered that the greater terms be kept upon the line AB; as was before prescribed in the first Chapter. CHAP. V. OF NAVIGATION. Some things in this kind will be performed very conveniently by these lines: As, SECT. 1. To make a Sea-chart after Mercators' projection. A Sea-chart may be made either general or particular; I call that a general Sea-chart, whose line AE, in the following figure, represents the equinoctial, as the line AE there doth the parallel of 50 gr. and so containeth all the parallels successively from the equinoctial towards either Pole: but they can never be extended very near the Pole because the distances of the parallels increase so much, as the Secants do. But notwithstanding this, it may be termed general, because that a more general Chart cannot be contrived in plano, except a true Projection of the Sphere itself. And I call that a particular Chart which is made properly for one particular Navigation, as if a man were to sail between the Latitude of 50 and 55 gr. and his difference of Longitude were not to exceed 6 gr. than a Chart made (as the figure following is) for such a Voyage, may be called particular. Now the making of such a Chart, is Master gunter's first proposition page 104 of the Sector, and this the line of Secants will sufficiently perform. For it were required to project such a Chart: Having drawn the line AB, and having crossed it at right angles with another line AE, representing the parallel of 50 gr. you must then take the Secant of 51 from your Scale, and set it from 50 to 51 on both sides the Chart, and draw the parallel 51 51. Again, take the Secant of 52 from your Scale, and set it upon your Chart from 51 to 52, and so draw the parallel 52 52. And so you are to draw the rest of the parallels. If therefore you take the Radius, and run it above and below, you shall make the spaces or distances of the Meridians such as in the bottom of the Chart are figured with 1, 2, 3, 4, 5, 6. These degrees thus set on the Chart, may be subdivided into equal parts, which in the graduations above and below ought so to be. But in the graduations upon the sides of the Chart, they ought as they go higher, still to grow greater. Yet the difference is so small that it cannot produce any considerable error, though the subdivisions be all equal. Divide them therefore either into 60 minutes, or English miles, or into 20 leagues, or into 100 parts of degrees, as shall best be liked of. It a little more curiosity should be stood upon for the graduations of the Meridian, instead of the Secants of 51, 52, 53, &c. you may take 50½, 51½, 52½, &c. always half a degree less than is the Latitude that should be put in. Now if each of those divisions at the bottom of the Chart, as A 1, &c. be made equal to the common Radius of the Sines, Secants, and Tangents, and if a Chart be made to that extent upon a skin of smooth Velame; well pasted on a board; you may work upon it many conclusions very exactly. The uses of the Sea-Chart Are set down in 12 Propositions by Master Gunter, beginning page 121. In each of which Propositions is showed how to resolve the Question upon the Chart itself, which will be direction enough how the work must be performed, without any more words here used. The working of these propositions also may be applied to the Scales of Sines and Tangents, on the Ruler, and wrought by protraction, according to the rules given in the first Chapter, if the proportions, as he lays them down in the forecited pages, be so applied. If a Scale of Rumbs be thought more expedient for these operations than is a Scale of Chords, it may be put into some spare place of the Ruler. His two Propositions, page 114. 116, may be done upon the Chart as is there showed, but his second Proposition, which is, SECT. 2. To find how many Leagues do answer to one degree of Longitude, in every several Latitude. THis (I say) may be done upon the Scales of Sines and equal parts: And for this purpose, the two last inches of the same Scale of equal parts, being equal in length to the Radius or Sine of 90, are divided into 20 at one end, and into 60 at the other end. Take therefore upon the line of Sines, the compliment of the parallels distance from the Equator, (or the compliment of the given Latitude) and measuring it upon the Scale of 20 parts, it will show you what number of Leagues make one degree of Longitude in that parallel of Latitude. And being measured upon the Scale of 60 parts, it gives so many of our miles, or so many minutes of the equinoctial, or any other great circle, as are answerable to one degree of Longitude in that Latitude. Example, Let it be required to find how many Leagues do answer to one degree of Longitude, in the Latitude of 18 gr. 12'. Take out of the line of Sines, the compliment of the given Latitude, namely. 71 gr. 48'. Then applying this distance to the Scale of 20 equal parts, you shall find it to reach 19, and so many Leagues do answer to one degree of Longitude, in the Latitude of 18 gr. 12'. And the same distance being measured upon the Scale of 60 equal parts, will give you 57 parts, and so many minutes of the Equator are answerable to one degree of Longitude, in that parallel of Latitude. So likewise, in the Latitude of 25 gr. 15', if you take the compliment thereof 64 gr. 45', out of the Scale of Sines, and apply it to the former line of 20, you shall find it to reach 18 parts, and so many Leagues do answer to one degree of Longitude, in the Latitude of 25 gr. 15'. ¶ In the Appendix to Master Norwood's Doctrine of Triangles, there is by him laid down 15 Questions of sailing by the plain Sea-chart, and others by Mercators' Chart, all which the line of Chords and equal parts will sufficiently perform, if the work of the third Chapter of this book be rightly understood. SECT. 3. How to set any place upon your Chart, according to its Longitude and Latitude. IF the two places lie under one parallel, and so differ only in Longitude, than the course leading from one to the other is East or West: As A and E being two places under the parallel of 50 gr. and differing 5½ gr. in Longitude. But if the two places differ only in Latitude, and lie under one Meridian, as A and B, then the course is North or South. CHAP. VI. Of Projections of the Sphere. FOr this purpose chiefly, is the lesser line of Chords added, being made to the same Radius that belongs to the Sines, Secants, and Tangents. For when any Projection is to be made, the fundamental Circle must be of that common Radius, and then the angles to be inserted upon it may be taken out of this line of Chords which is fitted to it. See the second Book of the Sector, Chap. 3. For these Tangents and Secants will perform the same things in those Stereographicke projections that there are done; and in all other irregular projections likewise. By this kind of work may any spherical conclusions be performed by protraction in plano. Also true Schemes of the Sphere may be drawn, suitable to any question, which will not a little direct in spherical calculations. As suppose it were required to project the Sphere suitable to this Question. Having the Latitude of the place, the declination of the Sun, and the Altitude of the Sun, to find either the Azimuth or the hour of the day. First, With the Radius of the line of Chords, upon the centre C describe the fundamental Circle ZHNO representing the Meridian, and draw the diameter HO for the Horizon, and ZCN at right angles thereto, ZN being the Zenith and Nadir points. Then by your line of Chords set the Latitude of your place (which let be 51 gr. 32') from Z to E, and from N to Q, drawing the line ECQ for the equinoctial, and at right angles thereto, the line MP for the axis of the World, P representing the North, and M the South Pole Secondly, Supposing the Sun to have 20 gr. of North Declination, take 10 gr. (the Semitangent of the Declination) out of the line of Tangents, and set it from C to G. Likewise, take 20 gr. (the Declination) from your line of Chords, and set that distance upon the Meridian from E unto D, and from Q unto K: then describe the ark of a Circle which shall pass directly through the points DGK, the centre whereof will always fall in the line CP if it were extended, and this ark DGK shall be the line of the sun's course when his Declination is 20 gr. from the equinoctial Northward. ¶ 1. The centres of the parallels of declination, and of the parallels of altitude, may readily be fowd by the Scale itself; as in this projection, having found the point F upon the Line ZC, extend the line ZC without the circle, and because the sun's altitude is 50 gr. take therefore out of your Scale the Secant of 40 gr. (the compliment of the Altitude) and set that distance from C to I, so shall the point I be the centre of the parallel of Altitude. Or take the Tangent of 40 gr. out of your Scale and place it from F to I, either of which will fall in the point I, the centre of the parallel of 50 gr. In the same manner may the centre of the parallel of Declination be found, by taking out of your Scale the Secant compliment of the Declination, and setting it from C, upon the line CP, (being extended) and where that distance ends, that is the centre of the parallel. ¶ 2. For the finding of the centres of the Hours and Azimuths, the Scales of Secants and Tangents will much help you; So the Azimuth from the South being 49 gr. 52', if you take the Tangent thereof out of the Scale of Tangents, & set it upon the Horizon from C to L: the point L shall be the centre of the circle NTZ. Or the Secant of 49 gr. 52' being set from T, that also shall give you the centre 'las before. The centre of the hour-circle is found in the same manner, for the hour from the Meridian being 31 gr. 28', if you take the Tangent thereof our of your Scale, it shall reach from C to R, the point R being the centre of the hour-circle musp. Or the Secant of 31 gr. 28', being set from V, shall give the point R for the centre of the same hour. And in this manner may any hour or Azimuth whatsoever be drawn. Many other propositions in astronomy, may be wrought upon this projection, and indeed any of the 28 cales of spherical Trigonometry, may by this kind of projection be easily illustrated and resolved, which will clearly inform the fancy in the resolving of spherical Triangles. An Example or two for practice shall be, 1. To find the sun's Amplitude. In this projection, the Amplitude from the East or West is represented by the line CX, take therefore the distance CX in your compasses, and apply it to the line of Tangents, (counting every degree of the Tangents to be two degrees) and where it resteth, that shall be the Amplitude from the East or West, which will be found to be 33 gr. 22x. Or if you lay a Ruler upon Z and X, it will cut the Circle in Y, and the distance NY being measured on the line of Chords, shall give the Amplitude also. 2. To find the distance of the Sun from the Zenith. The distance of the Sun from the Zenith is the ark Z S, therefore to find the quantity thereof, you must first find the pole of the circle NS Z, which is done after this manner. Lay a Ruler from Z to T, and it will cut the circle in a, then take in your compasses a quadrant of the outward circle, and set it from a to b, then lay a ruler from Z to b, and it shall cut the Horizon in e, which point e is the pole of the circle ZTN. Now to measure the ark Z S, you must lay a ruler upon e and S; which will cut the outward circle in the point A, so shall A Z, being measured upon the line of Chords, give you the quantity of degrees contained in the ark Z S, which will be 40, equal to the compliment of the sun's Altitude. ¶ This latter proposition was inserted rather to show how the ark of any great Circle of the Sphere (the sides of all spherical Triangles being such) may be measured, then for any need there was to find the distance of the Sun from the Zenith, for that mighe have been more easily effected, it being only the compliment of the sun's Altitude; but according to this operation, may the side of any spherical Triangle whatsoever be measured. The line of Sines also will project the Analemma, as Master Gunter showeth, if this proposition be added. How to divide any line given, into such parts as the Scale of Sines is divided. Which proposition may be done by that which is set down in the 1 Chapt. For if AD mn C were parts or divisions made equal to those upon the Scale of Sines, and CE were a line in the same manner to be divided: After you have prepared your work as is there prescribed, you need only to take the least distances between the points C nm D and the line AE, and insert the same into your given line, so shall the divisions thereof be proportional to the line of Sines. CHAP. VII. Of the line of Versed-Sines. THe general use of this Scale is principally to resolve these two spherical Cases. First, By having three sides of a spherical Triangle, to find an angle. Secondly, By having two sides and the angle comprehended, to find the third side. According to which two general cases you shall find particular examples; namely, the first and third Sections of this Chapter suitable to the first Case: and the 5 Section answerable to the second. SECT. 1. To find the sun's Azimuth. FIrst, find the sum and difference of the compliment of your Latitude, and compliment of the sun's altitude. Then having made AB equal to the length of the whole Scale, count upon the same Scale the sum and difference before found. After this, take with your Compasses the distance from the sun's place to the sum, and setting one foot of that extent upon B, with the other describe the ark CD. So again, take the distance upon the Scale from the sun's place to the difference, and with that extent upon the centre A, describe the ark EF: Which done, draw the straight line DE, so as it may justly touch those two arks, cutting the line AB in G: so shall BG (being measured upon the Scale, from the beginning of it) show the Azimuth from the South. And AG measured upon the same Scale will give the Azimuth from the North. SECT. 2. To find the Amplitude of the Suns Rising or Setting. IF you suppose the Sun to be in the Horizon, or 00 gr. high, and so the compliment of the Altitude to be 90, and if (upon these suppositions) you work as in the last Section is showed, then shall BG give the gradual distance of the Suns rising or setting from the South; AG from the North, and from the midst of the line to G, is the Amplitude from East or West. SECT. 3. To find the hour of the Day. MAke AB equal to the whole Scale, as before: and count from the beginning of the Scale to the sun's place what number of degrees there are; the same number shows the gradual distance of the Sun from the North Pole. Of this distance and the compliment of your Latitude, find the sum and difference, and count them both upon the Scale, as was done before. Then again, count thereon also the compliment of the sun's altitude: Upon which point, setting one foot of your Compasses, extend the other to the forenamed sum; and with that extent upon the centre B describe the ark CD. Again, setting one foot of your Compasses upon the compliment of the sun's altitude, extend the other to the forenamed difference, and with that extent upon the centre A, describe the ark EF. Lastly, draw the straight line DE, which only touching the two former arks, may cut the line AB in G: so shall AG (measured on the Scale, from the beginning of it) give the degrees of the sun's distance from the South. These may be turned into hours, counting 15 gr. for one hour, and 1 gr. for 4 minutes of an hour. SECT. 4. To find the semidiurnal and seminocturnal arks. IF you suppose the sun's altitude to be 00 gr. and so the compliment of it to be 90, and then work as is directed in the 3. Sect. of this Chap. then shall AG give the semidiurnal ark, and BG the seminocturnal ark: Each of these turned into hours and minutes, and doubled, will give the length of the Day and Night. SECT. 5. The sun's place being assigned in any point of the ecliptic, to find his Altitude at all hours. BY this, may Tables of the sun's Altitude be made to all hours, the Sun being in any sign of the zodiac, whereby many particular Instruments for finding the hour of the day, may be made, as Rings, Quadrants, Cylinders▪ and such like. [This always, and then only, happens, when the sum (found at first) is greater than 90 gr.] Look then how much it is beyond, for so many degrees is the Sun below the Horizon at that hour of the night: Or (which is all one) so many degrees is the Sun elevated above the Horizon in that sign or point of the ecliptic which is so much on the other part of the equinoctial. That is, If the sun's place given were the beginning of Taurus or Virgo, and your Compasses (suppose at the 9th. hour) go beyond the 90th gr. of the Scale, you shall there see how low the Sun is under the Horizon at 9 a clock at night, or at 3 in the morning. And the same also showeth how high the Sun is at 9 in the morning, or at 3 afternoon, if his place were in the entrance of Scorpio or Pisces, which two signs are so much beyond the equinoctial on the other part, as Taurus and Virgo are on this side. SECT. 6. All Proportions in Sines alone, where the Radius stands first, may be wrought upon this Scale, without any protraction at all. THe manner of the work will best appear by an Example. Let the proportion set down before in Sines alone be here repeated. The terms stand thus: As the Radius, to the Sine of 23½; So the Sine of 10, to the Sine of what? Take the sum and difference of the second and third arks, the sum is 33½ the difference is 13½: count these both upon the Scale, and there take their distance: apply the same to the middle of the Scale; so as that the same number of degrees may be above 90, that is below; so shall the degrees either above or below, be about 4; and this is the Sine required for a fourth proportional to the former. CHAP. VIII. How to work proportions in Sines and Tangents, by the lines of Versed-Sines and ⊙. Let the Sines (given or required) be measured out of ⊙, and let them be set upon the Radius from A, to AC or AE. Let the Tangents (given or required) be measured out of the Versed Scale, from 90 to 00, or to 180, which are 90 Chords belonging to 90 equal parts of the Semicircle ABDP, and the same Tangents must ever be set upon the Circle from A, as AB, AD: Then draw a right line through the first and third of the given terms, as from B and C to O; and another right line from O to D or to E. So the fourth term required shall be either the Sine AE, or the Tangent AD, each to be measured in its proper Scale. The further use of this line is shown afterwards in the making of declining reclining Dials. CHAP. ix.. To find the declination of a Plain. TO effect this, there are required two observations: the first is of the horizontal distance of the Sun from the pole of the plain, the second is of the sun's Altitude, thereby to get the Azimuth. And these two observations must be made at one instant of time, as near as may be, that the parts of the work may agree together the better. 1 For the horizontal distance of the Sun from the pole of the plain: Apply one edge of a Quadrant to the plain, so that the other may be perpendicular to it, and the limb may be towards the Sun, and hold the whole Quadrant horizontal as near as you can conjecture: Then holding a thread and plummet at full liberty, so that the shadow of the thread may pass through the centre and limb of the Quadrant, observe then the degrees cut off by the shadow of the thread, and number them from that side of the Quadrant that standeth square to the plain, for those degrees are the distance required. 2 At the same instant observe the Altitude of the Sun, these two will help you to the plains declination by the rules following. First, By having the Altitude, you may find the Azimuth by the 1. Sect. of the 8. Chap. then by comparing the Azimuth and distance together, you may find the plains declination in this manner. When you make your observation of the sun's horizontal distance, mark whether the shadow of the thread fall between the South and that side of the Quadrant which is perpendicular to the plain. For, 1. If the shadow fall between them, than the distance and Azimuth added together, do make the declination of the plain, and in this case, the declination is upon the same coast whereon the sun's Azimuth is. 2. If the shadow fall not between them, than the difference of the distance and Azimuth is the declination of the plain, and if the Azimuth be the greater of the two, than the plain declineth to the same Coast whereon the Azimuth is: Otherwise, if the distance be the greater than the plain declineth to the contrary Coast to that whereon the sun's Azimuth is. ¶ Note here further, that the Declination so found is always accounted from the South, and that all Declinations are numbered from either South or North towards either East or West, and must not exceed 90 gr. 1. If therefore the number of declination exceed 90, you must take the residue of that number to 180 gr. and the same shall be the declination of the plain from the North. 2. If the number of declination do exceed 180 gr. than the excess above 180 gives the plains declination from the North, towards that Coast which is contrary to the Coast whereon the Sun is. ¶ And here note, that wheresoever in this Chapter the use of a Quadrant is required, the Scale of Chords will effect the same; if upon a piece of plain board you describe a Quadrant, whose sides may be parallel to the edges of the board, upon which you may set off the horizontal distance and Altitude, which will perform the work thereof when a Quadrant is not at hand. CHAP. X. OF DIALS. To draw upright declining Dials, by the lines of Sines and Tangents. THe declination of the plain being found by the last Chap. Upon your plain describe a rectangled parallelogram, in which let the sides AB and CD be perpendicular to the Horizon, and each of them equal to the Tangent of your Latitude: and let AC and BD be equal each of them to the cotangent of your Latitude, and let BD be prolonged if need be. Then taking that side of the parallelogram (for the hour of 12) which looketh towards that coast unto which the plain declineth, as here namely, the side AB; and on that line having assumed the superior angle A in South-declining-plains, or the inferior in North-decliners, for the centre of your dial: Let BE and CG be made equal to the Sine of the plains declination, so AE being drawn, shall be the substilar, and AG shall be the hour of 6. Then from E, raise EF perpendicular to AE, and make A 12 and EF equal to the cousin of the declination: and if you draw of, the same shall represent the Axis, and the angle FAE showeth how much the same is to be elevated above the substylar. Again, make AH equal to the cousin of your Latitude, and draw H6 parallel to AB; which will cut AG in the point noted with 6. To this A6, let A6 also beyond the centre be made equal, and then draw the lines 12 6 and 12 6, which lines must have the hour points set upon them; and to perform that work do thus. Draw upon paper, or some other plain, the line LM, upon which set LR and RM, each of them equal to your Tangent of 45 gr. Then make RN equal to the Tangent of 30, and RO equal to the Tangent of 15, so shall you have points to find all the hours, and if you desire halves and quarters, you must also put their Tangents into the same line RM. Being thus prepared, if you would divide the lesser line 12 6 into its requisite parts, take the same line in your Compasses, and with it, upon L as a centre describe the ark PQ, and from M draw MP, which may only touch the same ark. Then from N take the least distance to the line MP, and the same will reach from 12 to 11, and from 6 to 7; so the least distance from O to the line MP, will give from 12 to 10, and from 6 to 8. And the least distance from R will reach from 12 or 6 to 9 In the same manner you must divide the larger line 12 6. Take it out of your dial, and with it describe the ark ST from the same centre L, and draw MS touching only the same ark. Then the least distances from N, O, R, to the line MS, will give the points or distances 12 1; 6 5; and 12 2; 6 4; and 12 3; or 6 3. These upon the South-decliner; the like may be done upon the North-decliner. Lastly, from the centre A, through these points you must draw the hourlines. An upright plain declining from the South towards the East, 30 gr. An upright plain declining from the North towards the East, 30 gr. In both these Plains. a rectangle parallelog. AB= tang. Lat. BD= cotang lat. BE= CG= sine declination. A12= EF= cousin declina. AH= cousin lat. A6= A6. RL= RM= tang. 45. RN= tang. 30. RO= tang. 15. CHAP. XI. Of the horizontal and full South Dials. The upright North plain is the same with the South, only turned upside down, and the course of the figures altered. The East and West upright plains may be made by the Tangent line, in such manner as others have prescribed. CHAP. XII. Of the Scale of horizontal Spaces. FOr the horizontal plains in special, there is a peculiar Scale by which the hours may suddenly be described, to any Latitude between 30 and 60 degrees. The manner of which work is easy. For you have the numbers from 30 to 60 five times repeated, serving for the five hours in so many Latitudes. Suppose then a horizontal dial were to be described for the Latitude of 51½ gr. First, by the Radius (which is from the beginning of the line to R) describe a Circle, and draw the line of 12 from the centre. Then take from the beginning of the line to VI, and set it in the Circle both ways from 12, these two are the points of the two sixes. Again out of the same Scale take the length from the beginning to 51½ in the remotest numbers, and set that upon the Circle on both sides 12, these are the points of 5 and 7. So from the beginning of the Scale to 51½ in the next remotest numbers, being set as the other were, will give the points of 4 and 8. The third 51½ will give the points of 3 and 9 And the fourth gives 2 and ●0. The last gives 1 and 11. ¶ The Chord line that is fitted to this horizontal Scale, is of good use in other delineations: But the further use of these two jointly, must be referred to another place. CHAP. XIII. How to draw upright declining Dials when the Latitude of the place is very little or very great. IN the work of the 10 Chap. it may fall out that either the Tangent or cotangent of the Latitude may be too great, such as the Scale will not afford. This will frequently fall out in the new Latitude of reincliners: to remedy that inconvenience, I have added these helps. Where the Latitude is but small A rectangle parallelo. 1. AB= CD= tang. of Latit. 2. BD= AC= Radius. 3. BE= CG= Sine declination. 4. A12= EF= cousin of declination. 5. AH= cosine of Latitude. 6. H m ♒ AB. 7. Draw EG, it will cut CD in K. AK is the line of six: it cuts H m at 6, make A6= A6, on both sides, and draw 12 6; 12 6; and divide them as the other are in the 10 Chap. Or you may draw BC the diagonal, and eke ♒ thereto, and so omit CG. Or you may make the ∠ DEK= to your Latitude, and so omit the two former. Or thus. After the 1, 2, 3, 4, 5, you may omit the 6. Then 7thly. Draw EG it will cut CD in K, and AK is the line of six. Then lay a Ruler from 12 to H, cutting DC in L. Make 12M= CL, and AN= AK. So shall KM, MN, be ♒ to the two former lines 6 12; 6 12; and may supply their Offices somewhat better, because they are larger. Where the Latitude is great. a rectangle parallelogr. 1. AB= CD= Radius, 2. BD= AC= cotang lat. 3. BR= CG= Sine declination. GR a right line cutting DB in E. AE Substilar. AG hour of 6. 4. GP= RT= A12= cousin of declination. TP a right line, cutting BD in O. AEF a right angle. 5. EF= EO. of the Style. 6. AH= cousin Latitude. him ♒ AB, cuts AG in 6. A6= A6, on both sides. Draw 6 12; 6 12, &c. Or after the 1, 2, 3, 4, 5. 6. Draw 12 H, it cuts DC in L. Make 12 M= LC: and AN= AG. Then GM, MN shall be ♒ to 6 12; 6 12: and may therefore supply their uses. CHAP. XIIII. Concerning Reclining and Inclining Plains, how to draw hours upon them. THey may be referred to a new Latitude, in which they shall stand as upright plains: and then the delineation will be the same with those in the 10 Chap. The Meridian line is not here to be taken for the line of 12 at midday (for it often represents the midnight) but for that part which helps to describe the dial. 1. The first thing to be done upon these plains, is (by some level) to draw the horizontal, and then the vertical line perpendicular thereto. 2. Next is the placing of the Meridian upon the plain, in a true position. In direct plains that re/ in-cline, and in upright decliners, the Meridian is the same with the plains vertical line. In East and West re/ incliners, it is the same with the horizontal line. In the rest, it ascendeth or descendeth from the horizontal line, and must be placed according to the rules hereafter given. I. ZB re|in-clination. BG ♒ ZO OC cousin of declination. RC ♒ FO. ORD Radius. DS ♒ boy. Out of this Structure will follow. DAE new Latitude less than 90. OR cousin of new Declination. DS Sine of Meridians ascension or descension. II. ZB re|in-clination. BG ♒ ZO. OA Sine of declination. HAI ♒ FO. IO Radius. AM ♒ IO. Out of this Structure will follow. OK is the sine of ZD or ND in the former figure, where the new Latitude DAE will be found. HO Sine of new declination. AM cousin of Meridians ascension or descension. ¶ How all re-inclining plains (being counted as upright in their new Latitude) are to be taken; whether as North or South decliners. As also, ¶ How the Meridian line is to be placed, whether ascending above, or descending below the horizontal line: and from which end of that line, whether that which looks the same way with the declination of the plain, or that which looks the contrary way. In North re/ incliners. If D fall below P, the Recliners are North plains, and the Meridian ascends above the horizontal line, from that end of it which looks to the same Coast of declination. Incliners are South plains, and the Meridian descends below that end of the horizontal line, which looks to the contrary Coast of declination. If D fall above P, the Recliners are South plains, and the Meridian goes below: contrary, Incliners are North plains, and the Meridian goes above the end looking the same way with declination. In South re/ incliners If D fall above AE, the Recliners are North plains, and the Meridian goes above the horizontal line, from the same end with the Coast of declination. Incliners are South plains, and the Meridian goes below the horizontal line, from that end which is contrary to the Coast of declination. If D fall below AE, the Recliners are North plains, and the Meridian goes below the horizontal line: contrary, Incliners are South plains, and the Meridian goes above the horizontal line, from that end which looks to the Coast of declination. If D fall into P, both re/ incliners, are called Polar plains, and the Meridian, in both, ascends from the Same end in Recliners. contrary end in Incliners. If D fall into E, the Recliners are North plains, and the Meridian ascends from the same; descends from the contrary end to that which looks upon the Coast of declination. Incliners are South plains, and the Meridian ascends from the contrary; descends from the same end that looks upon the Coast of declination. ¶ East & West Recliners are North plains, declining from North, So much as the compliment of their re/ in-clination comes to. This is their new declination, & their new Latitude is the compliment of the Latitude of your place. Incliners are South plains, declining from South, 4. For that which follows, take notice of these four things. First, That from D to the nearest AE (measured by the line of Chords) gives the new Latitude, in which the re/ in-clining plain, is an upright declining plain. Secondly, That OR (measured upon the line of Sines) gives the compliment of the plains new declination in that new Latitude: this New declination is to the same Coast with the Old, but always less in quantity than it. Thirdly, That DS (measured upon the Sines) gives the quantity of the Meridians ascension or descension. This gives the quantity, the former rules gave the Coast. Fourthly, That in the description of the dial, you must only make use of the new Latitude, and new Declination: having nothing to do with the other. 5. Having the former things known, you must (by the Tangent and cotangent of the new Latitude) describe your Rectangled Parallelogram (as in the 10 Chap.) and according as the plain was discovered to be a decliner from the North or South, you must make choice of your centre, place the substylar, style, and six a clock line, by help of the Sine and cousin of the new declination, and new Latitude, and then prick down and draw the hours, all in the same form that was before showed in the 10 Chap. for upright decliners. This for the dial's description. 6. Lastly, for placing your dial. First, Consider which way, and how much, your Meridian ascended or descended from the horizontal line. Then go to your plain, and there draw the same Meridian line answerably, setting off so many degrees by your Scale of Chords. When this is done, take your paper description, and lay the Meridian of it, either upon, or else parallel to, the Meridian drawn upon the plain, and take care to place it the right way; namely so, as that the imaginary style of your paper (or a real pattern of the style cut fit and set upon the paper dial) may point into the North or South Pole, according as the plain is esteemed to be a North or South plain. After this is performed, you may transfer each hour from the paper to the plain, and so finish all the work. CHAP. XV. Concerning full East and West reinclining plains. HEre in this sort of plain, you are only to take notice, that the new Latitude (wherein they stand as erect plains) is ever the compliment of your own Latitude. And the new declination (in that Latitude) is the compliment of their re/ in-clination. By knowing these, you may describe the dial according to the 10 Chap. The Meridian line (in all these) lieth in (or parallel to) the horizontal line. All which things will appear also out of the former figures, if according to them you should make a draught, and suppose your plain to decline 90 degrees, as all these East and West plains do. All other things will follow of themselves, agreeable to other plains. CHAP. XVI. Concerning reincliners, that are direct, or have no declination. IF the line CB be placed (as is prescribed in the former figure) and drawn quite through, it will represent your plain that is re/ in-clining towards the North, and without any declination. So also BL, if it be drawn quite through, will represent such plains as re/ in-cline towards the South, and have no declination. For which lines so drawn (or imagined only) you may gather (according to the former rules) which of the Poles (A or X) is elevated, and how much it is elevated (which is showed by the ark CA or LX.) You may also see which end of the Meridian is to be taken for the substilar line, over which (in these direct plains) the stile is ever to be erected, and must stand. Then for drawing the hours, you have no more to do, but to describe an horizontal dial to that elevation, which is due to the plain. The manner whereof is showed before in the 11 Chap. CHAP. XVII. How to deal with those plains, where the Pole is but of small elevation. SUch plains whose styles lie low, cannot have the hourlines distinctly severed, unless the centre of the dial be cast out of the plain. In such cases therefore the dial is to be made without a centre, in this manner. 1. Place AB the Meridian, A● the substilar, of the style, by the rules before given in the 10 and 13 Chapt. omitting what is done for the line of six, being here of no consequence. 2. Find the plains difference of Longitude by the 18 Chap. following. 3. Assume any two points in the substilar AE, as at R and S, and through them draw two infinite right lines, at right angles to AE. 4. To the style of, draw the parallel GH, at any convenient distance, such as you shall think fit, for your new style to stand from your plain. 5. Take the least distance from R to GH, and set it upon the substilar from R to K. So from S to GH, set from S to L. 6. Upon the two centres K and L, describe two Circles: And in them both, make the two angles RKM, SLM, equal to the plains difference of Longitude; and set it on that side the substilar RS, upon which the Meridian AB standeth. 7. The rest of the work will be easy to finish, if you begin (in each circle) from the points at M, to divide them into 24 equal hours; and from the centres to those equal divisions, draw out lines to cut their respective contingent lines in 12, 11, 10, &c. And from each correspondent hour, you must draw the lines 12 12, 11 11, 10 10, &c. An upright plain declining from South towards East 80 gr. Difference of Longitude 82d 08' South Pole elevated 6 12 CHAP. XVIII. Having the Latitude of the place, and the plains declination, to find the plains difference of Longitude. IT must be understood, that the plain is supposed (in this work) to be always erect; and that therefore for re/ in-clining plains, the Latitude and declination here mentioned is meant of the new Latitude and new declination. Two ways to do it. Make ABC a right angle. I. AB Sine of new Latitude. BC Tangent of new declination. BAC is the difference of the plains Longitude from your Meridian. Make DEF a right angle. II. DE Radius. EG Sine of new Latitude. EF cotangent of plains new declination. Draw GH parallel to DF. HDE is the compliment of the difference of Longitude. Or DHE is the difference it self. If this work be done for upright plains in your own Latitude, which will be needful in far decliners, than instead of the new Latitude and new Declination here mentioned, you are to use your own Latitude, and the upright plains Declination. The new Latitude and Declination are for reinclining plains. CHAP. XIX. Of Polar Plains, on which the Pole is not elevated at all. THose are called Polar plains, upon which neither of the two Poles is elevated at all, but the plain lies parallel to the Axis, such are the upright East and West: and in every declination from the South some one recliner: in every declination from the North some one Incliner. The new declination of all Polar plains is their difference of Longitude, in these you must work by the 10 and 14 Chap. to place AB the Meridian, AE the substilar; & for the style of, it hath no elevation from the substilar, but is the same with it. So that the work will be much like that in the 17 Chap. Make GH for the style, parallel to the substyle AE, at some convenient distance. Then assigning any point in the line AE, as S, through it draw an infinite right line perpendicular to AE. And take the least distance from S to GH, make SL equal thereto. Upon L describe a circle, and make SLM equal to the difference of Longitude, on the same Coast from SL unto which the plain declineth, or to the same Coast upon which the first line of 12 namely AB standeth. Then having found the hour points upon the line which passeth through S, namely, 6, 7, 8, 9, 10, &c. draw lines through them, all parallel to them substilar AESL. CHAP. XX. Another way to prick down the hourepoints, by the Tangent line on the Scale. LEt the first four Sections of the 17 Chap. be performed according to the directions there given. After them, you must gather the angles at the Pole, by help of the plains difference of Longitude in this manner. Let the former example serve. The difference of that plains Longitude will be 82 gr. 08'. Out of this, take the greatest number of some just hour; viz. 75 gr. The remainder is 7 gr. 8'. Having then set down the substile 00 00, as in the Margin, write this 7 gr. 8'. next under it, to which add 15 gr. continually, and you shall produce all the following numbers as you there see them. And note, that in this work 82 gr. 8'. the difference of Longitude will ever stand against the hour of 12, if you work right. Then take the first number 7 gr. 8'. out of 15 gr. the remainder is 7 gr. 52'; set this above the substyle, and to this number add continually 15 gr. (or one hour) the numbers will be produced such as you here see. gr. ' 3 52 52 4 37 52 5 22 52 6 7 52 Substyle 00 00 7 7 08 8 22 08 9 37 08 10 52 08 11 67 08 12 82 08 When this is done, draw a right line, therein assuming the point S or R. Then upon your Scale of Tangents, count the numbers 7 08, 22 08, &c. in the Table, and take them off from the same Scale, setting them severally from S to a, b, c, d, e. So again, upon the same Scale of Tangents count the other numbers, 7 52, 22 52, 37 52, &c. and take them off thence severally, and place them from Sat f, g, h, i. But because the tangents upon the Scale go but to 63 gr. 26', it must therefore here be showed how those that exceed that quantity may be supplied. Namely thus, Double the number of degrees and minutes, and from the sum take 90 gr. so shall the Tangent and Secant of the remaining ark (both of them put together) give the Tangent required. As if in the former example, it were required to find the Tangent of 67 gr. 8' noted upon the line by the length S e, we must do thus. The double of 67 gr. 8' is 134 gr. 16', from which taking 90 gr. the remainder will be 44 gr. 16'. Accordingly we must first take the Secant of 44 gr. 16', and set it from S to y; then take the Tangent of the same 44 gr. 16'. and set it also forward from y to e, so shall you have S e the whole Tangent of 67 gr. 8'. as is required. Thus do for any other which shall go beyond the Scale. CHAP. XXI. A second way for reincliners. TAke notice of these terms. 1. Vertical distance, is the distance of the plains pole from the Vertex or Zenith of the place. 2. Polar distance, is the distance of the plains pole from the North pole. Preparatory works. 1. Draw the horizontal line upon the plain, The ways how to effect these are showed otherwhere, and are here taken as known. and cross it at right angles with a vertical line. 2. Get the plains re/ in-clination, and consequently the distance of the plains Pole from the Zenith of the place: which is here called the vertical distance. 3. Get the plains declination, and always account how much it is from the North. For that is here called the angle of Declination. SECT. 1. By the Scale of Versed Sines, how to find the elevation of the Pole above the plain: and which Pole it is, whether North or South, that is elevated. First, find the sum and difference of The compliment of your Latitude, The plains vertical distance. And observe likewise, that If the point of your Compasses (Applied to the Scale) do fall just upon 90, then is your plain a meridional or Polar plain, and hath no pole elevated above it. If it fall short of 90 then is the North Pole elevated; and the elevation is so much as the point falls short of 90. If it fall beyond 90, so much as it falls beyond, so much is the South Pole elevated. SECT. 2. To find the plains difference of Longitude from the South part of your Meridian, and which way the said difference of Longitude is to be taken. First, find the sum and difference of The compliment of your Latitude, The forementioned polar distance. Then make AB equal to your whole line of Versed Sines. And upon your Scale count your difference now found, and the forementioned vertical distance, taking the distance of these two as they are numbered upon the Scale. With which length upon A, describe the arch CD. Take also upon the Scale, from the vertical distance to the forementioned sum, and with that length upon B, describe the ark EF. Then draw the line FC, so as to touch both these arks, cutting the line AB in G: so shall AG (being measured upon the Scale) give the plains difference of Longitude from the South, which is here required. ¶ This difference of Longitude is to be taken to the same Coast in the heavens unto which the plain declineth, and may afterwards, in the description of the dial, be easily accounted either from the South or North part of the Meridian, viz. so as that the said difference may be always lesso than 90 gr. SECT. 3. To find how much the Substilar (or plains proper Meridian) must lie from the vertical line of the plain, and which way. Frist, find the sum and difference of The Polar distance, The vertical distance. Then make AB equal to your whole Versed Scale. And on the same Scale, take the extent from the compliment of your Latitude to the difference now before found, with which length, upon A as the centre, describe the ark CD. Also upon the Scale, take from the compliment of your Latitude to the sum here before found, and with that length, upon the centre B, describe the ark EF. then draw the line FC, justly touching both these arks, and cutting the line AB in G, so shall AG (being applied to the Scale) give the quantity of the angle here required. According to this angle the substylar line must always stand off from the vertical line of the plain. Which way must the Substilar lie from the vertical line. If the plain hath the North Pole elevated upon it, then must the substilar always lie from the upper end of the vertical line towards the North Pole, so much as the angle was (in the last Section) found to be. If the South Pole be elevated, than the substilar lieth always from the lower end of the vertical line towards the same South Pole, according to the forenamed angle. If the plain be meridional (upon which neither of the Poles is elevated) than the substilar must do either, or both; these two: according to the angle before found. According to these Rules you may place the substilar line upon the plain in its true position requisite. SECT. 4. To draw the hourlines upon the plain. FIrst, consider by the first of these Sections, whether it is the North or South pole that is elevated upon your plain. If it be the North pole, then must the centre of your dial stand downward, and the style must point upward to the said North pole. But if the South pole be elevated, than the centre of the dial is to be set upward, and the style coming from thence must point downwards into the South Pole. Lastly, From the centre of the dial A or B, and through the said unequal parts, draw right lines. These last lines shall give you 12 of your hours required: And if you draw each of them quite through the centre, you shall have the whole number of 24, of which, you may take such as are suitable and necessary for your plain. When your paper dial is thus finished, you may transfer it to your plain, by laying the substilar upon (or parallel to) the substilar before placed upon the plain, and so insert all the hours from the paper to the plain. After all this, you may make the style to the angle of the Poles elevation, and fit it in according to its requisite place and position. ¶ Note, that because some of the hour points found in the Circle will happen so near to the centre of the dial that you cannot well draw the hourlines true; you may therefore help yourself by that direction which I have given in my geometrical way. [This geometrical way shall shortly be published by the Authors own copy, with his own Demonstrations of the whole work.] For drawing hours upon plains that have small elevations, and upon Polar plains, use the former directions. CHAP. XXII. A third way for all reinclining Dials. SECT. 1. To find a reinclining plains difference of Longitude from the South part of your Meridian: and how much the plains Meridian or (substyle) must lie from the vertical line of the plain. I. II. III. Compliment of your Latit. 38 30 K 38 30 K 38 30 K Plains vertical distance. 100 00 Z 60 00 Z 30 00 Z Their sum. 138 30 98 30 68 30 Their Difference. 61 30 21 30 8 30 Their half sum. 69. 15 R compl. 20 45 V 49. 15 R comp. 40. 45 V 34. 15 R compl. 55. 45 V Their half difference. 30. 45 S. compl. 59 15 X 10. 45 S compl. 79. 15 X 4. 15 S compl. 85. 45. X Plains declinat. from Sou. 50. 00 Y 140 00 Y 160 00 Y Describe a Circle with your common (or lesser) Scale of Chords. And out of the same Scale make A Y= plains declination from South. Out of the line ⊙ make A R= R, and A S= S, & draw Y R M and M S B and make A D= A B. Out of the same line ⊙ make A V= V, and A X= X, and draw Y V N, and N X C. ¶ Then if K be less than Z C A D is the differ. of Longitude required. and C B is the angle between the substile & the vertical line. ¶ But if K be greater than Z C A D is the forementioned angle. and C B is the difference of Longitude. These two arks C D and C B, must be measured from 90 in the line of Versed Sines, and look what number of degrees they there cut, the same must be accounted for their quantities. SECT. 2. To find the elevation of the Pole above the plain: and which of the Poles it is, whether North or South, that is elevated. MEasure A B upon the Versed Sines (from 90) as before: the compliment of that is E B. Measure also E C upon the same Scale, in the same manner. Count these quantities E B and E C (so found) upon the line ⊙, and set them from E, to F and G, and make Er= R (taking E r out of the Scale of Versed Sines from 90) Draw r F O, and O G P. Measure E P upon your Scale of Chords, it will there give you the polar distance. If E P fall to be 90, it is a meridional plain, and hath no Pole elevated. If it be less than 90, the compliment of it is the elevation of the North Pole. If it be greater than 90. the excess is the elevation of the South Pole. ¶ Note, that the three figures following have relation to the three Columns of the foregoing Table; and to these rules last delivered. Figure I. In this first Figure C A D 62 49 C B 34 13 E P 122 00 Figure II. In this second Figure C A D 110 29 C B 42 19 E P 36 28 Figure III. In this third Figure C A D 118 26 C B 44 56 E P 14 00 SECT. 3. Which way must the Substilar lie from the vertical line? THe Rules are the same with those before in the second way of Dialling, where the same Question is propounded. You may therefore have recourse to them. Or thus. Upon all plains whereon the North pole South pole is elevated; the substilar must lie from the upper end lower end of the vertical line towards the full North. South. For drawing the hours, and finishing the dial, you must do as is prescribed in the 4th. Sect. of the former second way. For, having placed the Substilar, and knowing the plains difference of Longitude, you are to use the same course here that was there given. It will be easy to do these things in plains that are upright, and have no rein-clination. Note in all these three ways of Dialling. All directions here given suppose you to be in the Northern Hemisphere of the world. If therefore you should be in the Southern Hemisphere, you may easily make these precepts serve there too, by only altering the name of North, Northen, &c. and South Southern, &c. one into the other. FINIS. This Scheme hath relation to the 16th. Chapter, page 54. Errata. Page 32, in the Diagram, for H P G, read H F G. Page 51, for, If D fall into E, read, If D fall into Ae. Page 54, line 16, for For, read From. In the dial Page 57, at the other end of the line M, place the letter K upon the Substilar. Page 62, line 19, for L P, read S P. In the Diagram page 76, the letter R is wanting, at the intersection of the line Y M, with the line A E.