Mathematical Instruments
Book VIII. Ch. III.

Of the Construction and Uses of Instruments, for drawing upon Dials the Arcs of the Signs, the Diurnal Arcs, the Babylonick and Italian Hours, the Almacanters, and the Meridians of principal Cities.

We now proceed to describe upon Dials certain Lines which the Shadow of the Extremity of the Style passes over, when the Sun enters into each of the 12 Signs of the Zodiack.

Of the Trigon of Signs.

The first Figure represents the Triangle or Trigon of Signs, made of Brass or any other solid Matter of a bigness at pleasure. The Construction of this is thus: First draw the Line ab, representing the Axis of the World, and ac perpendicular thereto, representing the Radius of the Equinoctial, and about the Point a describe the circular Arc dce at pleasure. This being done, reckon 23$$\frac{1}{2}$$ Deg. both ways from the Point c upon the said Arc, for the Sun’s greatest Declination, and draw the two Lines, ad, ae, for the Summer and Winter Tropicks; likewise draw the Line de, which will be bisected by the Radius of the Equinoctial in the Point o, about which, as a Center, draw a Circle, whose Circumference passes thro’ the Points d and e of the Tropicks, and divide the Circumference thereof in 12 equal Parts, beginning from the Point d. Then thro’ each Point of Division equally distant from d and e, draw occult Lines parallel to the Radius of the Equinoctial Circle. These Lines will intersect the Arc dc in the Points thro’ which and the Center a Lines being drawn, these Lines will represent the beginnings of the Signs of the Zodiack, at 30 Deg. distance from each other,

But to divide the Signs into every 10th or 5th Degree, you must divide the Circumference of the Circle into 36 or 72 equal Parts. After this, we denote the Characters of the Signs upon each Line, as appears per Figure. And when the Trigon is divided into every 10th or 5th Degree, we place the Letter of the Month to the first 10 Degrees of each Sign agreeing therewith.

But the Trigon of Signs may be readier made by the means of a Table of the Sun’s Declination; for having drawn the two Lines ab and ac at right Angles, lay the Center of a Protractor on the Point a, with its Limb towards the Point c; and keeping it fixed thus, count 23$$\frac{1}{2}$$ Deg. on both sides the Radius ac, for the Tropicks of ♋︎ and ♑︎, 20 Deg. 12 Min. for the beginnings of the Signs ♌︎, ♊︎, ♐︎ and ♒︎, and 11 Deg. 30 Min for ♉︎, ♍︎, ♏︎. and ♓︎. And in this manner we divide the Spaces for each Sign into every 10th or 5th Deg. by means of the following Table of the Sun’s Declination. Note, The Equinoctial Points of ♈︎ and ♎︎ are placed at the end of the Radius of the Equinoctial ac.

A Table of the Sun’s Declination for every Degree of the Ecliptick.
Degrees of the Ecliptick Signs Signs Signs Degrees of the Ecliptick
♈︎ ♎︎ ♉︎ ♏︎ ♊︎ ♐︎
D. M. D. M. D. M.
10241151202529
20481212203628
31121232204827
4136125321026
5201313211125
62231333212124
72471353213223
83111412214222
93351432215121
103581451220020
1142215922819
124451528221718
13591547222417
14532165223216
155551622223915
166191640224614
176421657225213
18751714225712
19728173023211
20750174723710
2181318323119
22835181623158
23858183423187
24920184923216
2594219323245
26104191823264
271026193223273
281047194623282
29119195923291
301130201223300
♓︎ ♍︎ ♒︎ ♌︎ ♑︎ ♋︎

By this Table we may know the Sun’s Declination and Distance from the Equinoctial Points each Day at Noon, in every Degree of the Signs of the Zodiack, the greatest Declination being 23 Deg. 30 Min. tho’ at present it is but about 23 Deg. 29 Min. but a Minute difference is of no consequence in the Use of Dials. The Degrees of the first Column to the Left-hand, are for the Signs set down upon the Top of the Table, and the Degrees in the last Column numbered upwards, are for the Signs set at the Bottom of the Table.

Of the Trigon of Diurnal Arcs.

The second Figure represents the Trigon of Diurnal and Nocturnal Arcs. These are drawn upon Sun-Dials by Curve-Lines, like the Arcs of the Signs, and by means of them the Shadow of the Style shews how many Hours the Sun is above the Horizon, in any given Day, that is, the Length of the Day, and consequently the Length of the Night too; for this is the Complement of that to 24 Hours.

The Trigon of Signs is the same for all Latitudes, since the Sun’s Declination is the same for all the Earth: but the Diurnal Arcs are different for every particular Latitude, and we draw as many of these Arcs upon a Dial, as there are Hours of Difference between the longest and shortest Days of the Year.

Now to construct the Trigon of Diurnal Arcs upon Brass or any other solid Matter, first draw the right Line RZ for the Radius of the Hour-Line of 12, or of the Equinoctial; and about the Point R, with any Opening of your Compasses taken at pleasure, describe the circular Arc TSV, and lay off both ways thereon from the Point S, two Arcs, each equal to the Complement of the Latitude. For Example, if the Latitude be 49 Deg. make the Arcs SV, and ST, of 41 Deg. each. This being done, draw the right Line TXV, and about the Point X, as a Center, describe the Circumference of a Circle TZVY, which divide into 48 equal Parts by dotted Lines, drawn parallel to the Radius of the Equinoctial RZ: then these Lines will cut the Diameter TXV in Points, thro’ which and the Point R, you may draw the Radius’s of the Hours. And since the longest Day at Paris is 16 Hours, and the shortest 8, you need but draw four Radius’s on one Side the Line RZ, and a like Number on the other Side.

Moreover, the Angles that all the Radius’s make at the Point R may be found Trigonometrically, by the following Analogy, viz. As Radius is to the Tangent Complement of the Latitude, So is the Tangent of the Difference between the Semidiurnal Arc at the time of the Equinox and the Arc proposed, to the Tangent of the Sun’s requisite Declination. For Example; Suppose it be required to draw upon the Trigon the diurnal Arc of 11 or 13 Hours, the Semidiurnal Arc is 5$$\frac{1}{2}$$ Hours, or 6$$\frac{1}{2}$$ Hours, and the Day of the Equinox the diurnal Arc is 12 Hours; and consequently the Semidiurnal Arc is 6 Hours, and the Difference is half an Hour: therefore Radius must be put for the first Term of the Analogy, the Tangent of 41 Deg. (viz. the Complement of the Latitude of Paris) for the second Term, and the Sine of 7 Deg. 30 Min. for the third Term. Now the fourth Term being found, the Sun’s Declination is 6 Deg. 28 Min. South, when the Day at Paris is 11 Hours long; and 6 Deg. 28 Min. North, when the Day is 13 Hours; and making three other Analogies, you will find that the Declination of the diurnal Arc of 10 Hours and 14 Hours, is 12 Deg. 41 Min. of 9 Hours and 15 Hours, 18 Deg. 25 Min. and of 8 Hours and 16 Hours, 23 Deg. 30 Min.

Of the Trigon with an Index.

The third Figure represents the Trigon of Signs put upon a Rule or Index A, in order to draw the Arcs of the Signs upon great Dials. The diurnal Arcs may be drawn likewise upon this Trigon; but the Arcs of the Signs and diurnal Arcs too must not be drawn upon one and the same Dial, for avoiding Confusion. In. the Center of the Index there is a little Hole thro’ which is put a Pin, that so the Instrument may turn about the Center of a Dial. The Trigon (Tides along the Index, and may be fixed in any part thereof by means of the Screw B. The Arcs of the Signs with their Characters are round about the Circumference, and there is a fine Thread fixed in the Center thereof, in order to extend over the Radii quite to the Hour-Lines of a Dial, as we shall by and by explain.

The fourth Figure represents one half of a horizontal Dial, having the Morning Hour-Lines to 12 o’Clock thereon, and the Equinoctial Line CD.

This being enough of the Dial, for explaining the Manner of drawing the Arcs of the Signs thereon, by means of Figure 5, which represents a Trigon of Signs drawn upon a Plate, on which the Hour-Lines of an horizontal Dial are adjusted in the following manner.

Take the Length of the Axis VR of the horizontal Dial between your Compasses, and lay it off on the Axis of the Trigon from O to C; after this, take the Distance from the Center V of the Dial to the Point C, wherein the Equinoctial Line cuts the Hour-Line of 12, and lay it off on the Trigon from C to a, and draw lightly the Line ca12, cutting all the seven Lines of the Trigon. This being done, take upon this Line the Distance from the Point c to the Intersection of the Summer Tropick, and lay it off from the Center V of the Dial on the Hour-Line of 12, and you will have one Point thro’ which the Summer Tropick must pass; likewise take the Distance from the Point c to the Intersection of the Parallel of if, and lay it off on the Flour-Line of 12, from the Center of the Dial, and you will have a Point on the said Hour-Line thro’ which the Parallel of ♊︎ must pass; likewise assume all the other Distances on the Trigon, and lay them off successively on the Hour-Line of 12 of the Dial, from the Center to the Point thro’ which the Winter Tropick passes, which must be the most distant from the Center of the Dial, and you will have the Points in the Hour-Line of 12 thro’ which each of the Parallels of the Signs must pass. And by proceeding in this manner with the other Hour-Lines, you will have Points in them thro’ which the Parallels of the Signs must pass. For Example, Assume on the Hour-Line of 11 of the Dial, the Distance from the Center thereof to the Point wherein the Equinoctial Line cuts it, and lay this Distance off upon the Trigon from c towards a, and draw the right Line C11; then take the Distances from the Point c to the Intersection of each of the Parallels of the Signs, and lay them off from the Center of the Dial, on the Hour-Line of 11, to the Points 22, &c. and those will be Points in the Hour-Line of 11, thro’ which the Parallels of the Signs must pass. Understand the same for others.

But because the Hour-Line of 6 is parallel to the Equinoctial Line, make this likewise parallel to the Radius of the Equinoctial a on the Trigon: and to prick down the Line for the Hour of Seven in the Evening, describe an Arc about the Point C, as a Center, from the Line for the Hour of 6 to that for the Hour of 5; and lay off that Arc on the other Side of the Line for the Hour of 6, and then you may draw the Hour-Line of 7, which will not meet the Summer Tropick. Finally, The Line for the Hour of 8 must make the same Angle with the Line of the Hour of 6, as the Line for the Hour of 4 does; but it is useless to draw this Line for the Latitude of 49 Deg. because this Line being parallel to the Tropick of ♋︎, cannot cut any one Radius to the Signs. Now the Points thro’ which the Arcs of the Signs must pass, being found on the Hour-Lines of the Dial, you must join all those that appertain to the same Sign with an even hand; and you will have the curved Arcs of the Signs, whose Characters must be marked upon the Dial, as per Figure. Note We sometimes set down the Names of the Months, and of some remarkable moveable Feasts upon the Dial. The Arcs of the Signs are drawn upon vertical Dials in this manner; but here the Winter Tropick must be nighest to the Center of the Dial, and the Summer Tropick furthest distant from it.

Fasten the Rule or Index to the Center of the Dial by a Pin, so that it may be turned and fixed upon any Hour-Line, as may be seen in Figure 6: then having fixed the Center of the Trigon upon the Index, at a Distance from the Center of the Index equal to the Distance from the Center of the Dial to the Extremity of the Axis thereof, by means of the Screw R; take the Thread on one Hand, and with the other raise or lower the Instrument upon the Plane of the Dial, so that the Thread extended along the Radius of the Equinoctial of the Trigon, meets the Point wherein some Hour-Line cuts the Equinoctial Line of the Dial, and in this Situation fix the Index. This being done, extend the Thread along the Radius’s of the Trigon, and prick down the Points upon each Hour-Line of the Dial, thro’ which the Parallels of the Signs must pass, both above and below the Equinoctial Line, as we have done on the Hour-Line of 12 of the Dial represented in Figure 6. And if you do thus on all the Hour-Lines successively one after the other, and the Points marked thereon appertaining to the same Sign, be joined by an even Hand, you will have the Parallels of the Signs upon the Surface of the Dial. But to make the Points on the Hour-Line of 6, the Instrument must be turned so that the Fiducial Line of the Index be upon the Hour-Line of 12, and the Radius of the Equinoctial Circle of the Trigon parallel to the Hour-Line of 6. The Instrument being thus fixed, extend the Thread along the Radius’s of the Signs, until it cuts the Hour-Line of 6, and the Points where it cuts the said Hour-Line, will be those thro’ which the Parallels of the Signs must pass in that Hour-Line.

When the Arcs of the Signs are drawn on one side of the Dial, for example, on the Morning Hour-Lines, you may lay off the same Distances from the Center on the Hour-Lines of the ether side the Meridian; as the Points denoted on the Hour-Line of 11 must be laid off on the Hour-Line of 1, those on the Hour-Line of 10 on the Hour-Line of 2; and so draw the Arcs of the Signs on the other side of the Meridian. Note, The Arcs of the Signs are drawn upon declining Dials in the same manner, if the Substylar Line be made use of instead of the Meridian, and the Distances from the Center be taken equal upon those Hour-Lines equally distant on both sides of the Substyle from it.

If the diurnal Arcs are to be pricked down upon a Dial instead of the Arcs of the Signs, that is, the Length of the Days, we may likewise put thereon the Hour of the Sun’s rising and setting, if the Length of the Day be divided into two equal Parts. For example, when the Day is 15 Hours long, the Sun sets half an Hour pass 7 in the Afternoon, and rises half an Hour past 4 in the Morning; and so of others.

If the Arcs of the Signs are to be drawn upon Equinoctial Dials, as on that of Figure 7, take the length of the Axis of the Style AD, and lay it off upon the Axis of the Trigon (of Figure 5) from O to P, and draw the Line PN parallel to the Radius of the Equinoctial; this shall cut the Summer Tropick and two other Parallels: then take the Distance from the Point P to the Intersection of the Tropick of ♋︎; and with that Distance about the Center A of the Dial draw a Circle, which shall represent the Tropick of ♋︎. Take likewise the two other Distances on the Parallel of the Trigon, and draw two other Circles about the Center of the Dial, the one for the Parallel of ♊︎ and ♌︎, and the other for that of ♉︎ and ♍︎, which may be drawn upon an upper Equinoctial Dial. But if this was an under Equinoctial Dial, then the above described Circles would represent the Parallels of ♏︎, ♐︎, ♑︎, ♒︎ and ♓︎: but as for the Parallels of ♈︎ and ♎︎, they cannot be drawn upon Equinoctial Dials, because when the Sun is in the Plane of the Celestial Equator, his Rays fall parallel to the Surfaces of Equinoctial Dials, and the Shadows of their Styles are indefinitely protended.

The Horizontal Line is thus drawn: First lay off the Style’s length on the Hour-Line of 6, and about the Extremity D thereof, describe the Arc EF (upwards for an upper Dial) equal to the Latitude, viz. 49 Deg. for Paris, and draw the Line DF, which shall cut the Meridian in the Point H, thro’ which the horizontal Line must be drawn parallel to the Hour-Line of 6, as may be seen in Figure 7.

The Use of this Line is to shew the rising and setting of the Sun at his entrance into the beginning of each Sign. For example, because it cus the Tropick of Cancer on the Dial, in Points thro’ which the Hour-Line of 4 in the Morning, and 8 in the Evening pass; therefore the Sun rises the Day of the Solstice at 4 in the Morning, and sets at 8 in the Evening at Paris. Understand the same of others.

To draw the Arcs of the Signs upon Polar Dials.

The Dial being drawn (as appears in Fig. 6.), the dotted Radii of the Hours continued out ’till they meet the Equinoctial Line must be laid off successively upon the Radius of the Equinoctial of the Trigon of Signs (Fig. 5.) for drawing as many Perpendiculars thereon as there are dotted Radii, viz. one for the Hour of 12, and the five others for the Hours of 1, 2, 3, 4, and 5, which shall cut the Radii of the Signs of the Trigon. This being done, take the Distances from the Radius of the Equinoctial of the Trigon upon the said Perpendiculars, to the Radius’s of the other Signs, and lay them off upon the Hour-Lines of the Dial on both sides the Equinoctial Line AB. For Example; Take the Distance 12♑︎, and lay it off on the Dial from the Point C upon the Hour-Line of 12, and you will have two Points in the said Line thro’ which the Tropicks must pass. Likewise take the Space on the Trigon upon the Line 5♑︎ or ♋︎, and lay it off upon the Hour-Lines of 5 and 7 on both sides the Equinoctial Line of your Dial, and you will have Points in the Hour-Lines of 5 and 7, thro’ which the Tropicks must pass. And in this manner may Points be found in the other Hour-Lines thro‘ which the said Tropicks must pass; as also the Points in the Hour-Lines thro’ which the Parallels of the other Signs must be drawn, which being found must be joined. Note, We have only drawn the two Tropicks in the Figure of this Dial for avoiding Confusion. And the Parallels of the Northern Signs must be drawn underneath the Equinoctial Line, and the Southern Signs above it. Also the diurnal Arcs are drawn in the same manner as the Arcs of the Signs are.

How to draw the Arcs of the Signs upon East and West Dials.

The Arcs of the Signs are drawn nearly in the same manner upon East and West Dials as upon Polar ones: for Example, Let it be required to draw the Arcs of the Signs upon the West Dial of Figure 8. the dotted Radii of the Hours produced to the Equinoctial Line CD, must be laid off upon the Trigon of Figure 1. from the Point a upon the Radius of the Equinoctial, that so Perpendiculars may be drawn upon the Trigon cutting the Radius’s of the Signs; after this, you must take upon the said Perpendiculars the Distances from the Radius of the Equinoctial to the Intersection of the Radii of the other Signs, and lay them off upon the Hour-Lines of the Dial, on both sides the Equinoctial Line. For Example, Take the Space 6♑︎, or ♋︎, and lay it off on both sides the Point D upon the Hour-Line of 6 on the Dial: Proceed in this manner for finding Points in the other Hour-Lines thro’ which the Curve Parallels of the Signs must be drawn with an even Hand, so that the Northern ones be under the Equinoctial Line, and the Southern ones above it. Note, The diurnal Arcs are drawn in the same manner; and we have only drawn the two Tropicks thereon for avoiding Confusion.

The Construction of a horizontal Dial, having the Italian and Babylonian Hours; as also the Almacanters and Meridians described upon it.

Having already shewed the manner of pricking down the Astronomical Hours upon Sun-Dials, as also the Diurnal Arcs, and Arcs of the Signs, there may yet be several other Circles of the Sphere represented upon Dials, being pleasant and useful, which the Shadow of the Extremity of the Style passes over; as the Italian and Babylonian Hours, the Azimuths, the Almacanters, and the Meridians of principal Cities.

The first Line of the Italian and Babylonian Hours is the Horizon, like as the first Line of the Astronomical Hours is the Meridian; for the Italians begin to reckon their Hours when the Center of the Sun touches the Horizon at his setting, and the Babylonians when he touches the Horizon at his rising.

A general Method for drawing the Italian and Babylonian Hours upon all kinds of Dials.

The Astronomical Hour-Lines, and the Equinoctial Line being drawn, as also a diurnal Arc or Parallel of the Sun’s rising for any Hour, at pleasure, as, for the Hour of 4 at Paris, which Arc will be the same as the Summer Tropick, you may find two Points (as we shall shew here) in each of the aforesaid Lines, viz. one in the Equinoctial Line, and the other in the diurnal Arc drawn, by means of which it will not be difficult to prick down the Italian and Babylonian Hour-Lines; because they being the common Sections of great Circles of the Sphere and a Dial-Plane, will be represented in right Lines thereon.

Now suppose it be required to draw the first Babylonian Hour-Line upon the horizontal Dial of Figure 7, first consider that when the Sun is in the Equinoctial he rises at 6, and at 7 he has been up just an Hour; whence it follows, that the first Babylonian Hour-Line must pass thro’ the Point wherein the Astronomical Hour-Line of 7 cuts the Equinoctial Line; the second thro’ the Intersection of the Hour-Line of 8; the third thro’ that of the Hour-Line of 9; and so of others.

But when the Sun rises at 4 in the Morning, the Point in the Tropick of ♋︎, wherein the Hour Line of 5 cuts it, is that thro’ which the first Babylonian Hour-Line must pass; the Intersection of the Hour-Line of 6 in the said Tropick, that thro’ which the second Babylonian Hour-Line must pass; the Intersection of the Hour-Line of 7 with the said Tropick, that Point thro’ which the third Babylonian Hour-Line must pass; and so of others. Then if a Ruler be laid to the Point wherein the Hour-Line of 5 cuts the Tropick of Cancer, and or the Point in the Equinoctial Line cut by the Hour-Line of 7, and you draw a right Line thro’ them; this Line will represent the first Babylonian Hour-Line. Proceeding in this manner for the other Babylonian Hour-Lines, you will find that the 8th Babylonian Hour-Line will pass thro’ the Point the Tropick of Cancer is cut by the Astronomical Hour-Line of 12, and the Point in the Equinoctial cut by the Hour-Line of 12; and the 5th Babylonian Hour-Line thro’ the Point in the said Tropick cut by the Hour-Line of 7 in the Evening, and the Point in the Equinoctial Line cut by the Hour-Line of 5.

One of the Babylonian Hour-Lines being drawn, it is afterwards easy to draw all the’ others; because they proceed orderly from one Astronomical Hour-Line to the other, on the Parallel and the Equinoctial Line, as appears per Figure. Finally, The Sun lets at the 16th Babylonian Hour, when the Day is 16 Hours long: he sets at the 12th when he is in the Equinoctial; and at the 8th when the Night is 16 Hours long, because he always rises at the 24th Babylonian Hour.

You must reason nearly in the same manner for pricking down the Italian Hour-Lines. Here we always reckon the Sun to set at the 24th Flour; and consequently in Summer, when the Nights are but 8 Hours long, he rises at the 8th Italian Hour; at the Time of the Equinox he rises at the 12th Italian Hour; and in Winter, when the Nights are 16 Hours long, he rises at the 16th Italian Hour: and therefore the Hour-Line of the 23d Italian Hour must pass thro’ the Intersection of the Astronomical Hour-Line of 7, and the Summer Tropick the Intersection of the Hour-Line of 5, and the Equinoctial Line, and the Intersection of the Hour-Line of 3, and the Winter Tropick. But two of the said Points are sufficient for drawing the said Italian Hour-Line. The 22d Italian Hour-Line passes thro’ the Intersection of the Hour-Line of 6 in the Evening, and Summer Tropick, the Intersection of the Hour-Line of 4, and the Equinoctial Line, and the Intersection of the Hour-Line of 2, and the Winter Tropick. Proceeding on thus, you will find that the 18th Italian Hour-Line passes thro’ the Points of the 12th Equinoctial Hour, that is, at the Time of the Equinox, it is Noon at the 18th Italian Hour; whereas at the Time of the Summer Solstice it is Noon at the 16th Italian Hour, and at the Winter Solstice it is Noon at the 20th Italian Hour, in all Places where the Pole is elevated 49 Degrees, as may be seen in the following Table.

A Table for drawing the Babylonian Hour-Lines upon Dials.
Babylonian Hours Passing in the Parallel of
♋︎ ♈︎ ♑︎
thro’
1579
26810
37911
481012
59111
610122
71113
81224
9135
10246
11357
12468
13579
146810
157911
1681012
A Table for drawing the Italian Hour-Lines upon Dials.
Italian Hours Passing in the Parallel of
♋︎ ♈︎ ♑︎
thro’
23753
22642
21531
204212
193111
1821210
171119
1612108
151197
141086
13975
12864
11753
10642
9531
84212

The Use of the Italian Hour-Lines upon a Dial may be to find the Time of the Sun’s setting, in substracting the Italian Hour present from 24; and by the Babylonian Hours may be known the Time of the Sun’s rising.

How to draw Almacanters, and the Azimuths.

The Almacanters or Circles of Altitude are represented upon the horizontal Dial by corncentrick Circles, and the Azimuths by right Lines terminating at the Foot of the Style B, which represents the Zenith, and is the common Center of all the Almacanters: and therefore you need but divide the Meridian BXII into Degrees, the Extremity of the Style C being the Center; and the Tangents of those Degrees on the Meridian will be the Semidiameters of the Almacanters, which shall terminate at the two Tropicks. Now to find these Tangents, you may use a Quadrant like that of Figure 8, in this manner: Lay off the Length of the Style CB from A to H, and draw the Line HI parallel to the Side AC of the Quadrant; then will this Line be divided into a Line of Tangents by Radii drawn from the Center A to the Degrees of the Limb. And these. Tangents may be taken between your Compasses, and laid off upon the Meridian Line BXII. in such manner, that the 90th Degree answers to the Point B. But since this Dial is made for the Latitude of 49 Deg. and so consequently the Sun in his greatest Altitude there, is but 64 Deg. 30 Min. you need only prick down this greatest Altitude, which will terminate at the Summer Tropick.

This being done, if one of the Circles of Altitude be divided into every 10th Deg. beginning from the Meridian BXII. which is the 90th Azimuth, and thro’ these Points of Division right Lines are drawn to the Foot of the Style B: these right Lines will represent the Azimuths or vertical Circles. We have not drawn them upon the Dial, for avoiding Confusion, but they may be easily conceived.

Now the Use of the Almacanters is to shew the Sun’s Altitude above the Horizon at any time, and of the Azimuths, to shew what Azimuth or vertical Circle the Sun is in: and this is known by observing what Circle of Altitude or Azimuth Line, the Shadow of the Extremity of the Style of the Dial falls upon.

How to draw the Meridians or Circles of Terrestrial Longitude upon the horizontal Dial.

About the Point D, the Center of the Equinoctial Circle, describe the Circumference of a Circle, and divide it into 360 equal Parts or Degrees, or only into 36 Parts, for every 10th Degree 5 then from the Hour-Line of 12, which represents the Meridian of the Place for which the Dial is made, viz. Paris, count 20 Deg. West ward for its Longitude, or Distance from the first Meridian passing thro’ the Point G; on which having wrote the Number 360, prolong the Line GD to E, in the Equinoctial Line, and afterwards from the Center A draw the first Meridian thro’ E, which passes thro’ the Island de Fer, and so of others. But it will be easier to draw the Meridians eastwardly for every 5th or 10th Degree, and place those principal Cities upon them whose Longitudes you know: as, for example, Rome is 10$$\frac{1}{2}$$ Deg. more eastwardly than Paris, Vienna 15 Deg. more eastwardly than the said City of Paris, and so of other eminent Cities, whose Differences of Meridians from that of Paris, are known by a good Globe, or Map, made according to the exact Observations of the Academy of the Sciences.

The Use of these Meridians on the Dial, is, to tell at any time when the Sun shines thereon, what Hour then it is under any one of the said Meridians, in adding to the time of Day at Paris, (for which the Dial is made) as many Hours as there are times 15 Deg. of Difference between the Meridians, and 4 Min. of an Hour for every Degree.

For example; When it is Noon by this Dial at Paris, it will be One a clock at Vienna, because this City is more to the East than Paris by 15 Deg. and consequently receives the Sun’s Light sooner than Paris does. And at Rome it will be 42 Min. past 12, because it is 10 $$\frac{1}{4}$$ Deg. more eastward than Paris, and so of others. These Lines of Longitude represent the Meridians of the Places attributed to them; so that when the Shadow of the Style falls upon any one of them, it will be Noon under that Meridian.