Of Regular and Irregular Dials, drawn upon Planes and Bodies of different Figures.

This Instrument represents a hollow Body, having 14 Planes, upon each of which a Dial may be drawn.

The upper Plane A, is parallel to the Horizon; and so upon this a Horizontal-Dial is drawn, as well as upon the under Plane E, whereon the Sun mines but a very little. The Plane B is parallel to the Axis of the World, and makes an Angle of 49 Degrees with the Horizon of Paris; for the Latitude of which, all the Dials are supposed to be drawn. Now upon this Plane is drawn an upper Polar Dial, and upon the Plane F, which is opposite thereto, is drawn an under Polar Dial. The Plane C is parallel to the Prime Vertical, and since it faces the South, there is drawn thereon a South Vertical Dial, And upon the opposite Plane to this, which is towards G, and faces directly to the North, is drawn a Vertical North Dial, which cannot be represented in this Figure.

The Plane H, which is parallel to the Equinoctial, and so makes an Angle with the Horizon of 41 Deg. viz. the Complement of the Latitude of Paris, hath an upper Equinoctial Dial drawn upon it; and upon the opposite Plane D, is drawn an under Equinoctial Dial. The Plane K is parallel to the Plane of the Meridian, and because it directly faces the West, a Meridional West Dial is drawn thereon, and upon the opposite Plane to this is drawn a Meridional East Dial. The Plane I makes an Angle 45 Deg. with the Meridian; and therefore there is drawn upon it a vertical Decliner, declining Southwestwardly 45 Deg. and upon the opposite Plane to this is drawn a North-East Decliner of 45 Deg. Finally, The Plane L declines North-West 45 Deg. and its Opposite 45 Deg. South-East; and so upon these two Planes are drawn North-West and South-East Decliners.

The first Nine of the abovementioned Dials, are called Regular ones; and the Four others, which decline, are called Irregular Dials.

The Axes of all these Dials are parallel to each other, and to the Axis of the World. We shall hereafter give the Construction of all these Dials, as well as of those on the following Instrument, of which we are going to speak.

This Figure is one of the five Regular Bodies, of which we have spoken in the first Book. This Body is called a Dodecahedron, and is terminated by 12 equal Pentagons, upon every of which may be drawn a Dial, except on the undermost.

The Plane A being Horizontal, hath a Horizontal-Dial drawn thereon, whose Hour-Line of 12 bisects one of the Angles of the Pentagon. Upon the Plane B, which faces the South, is drawn a direct South-Dial, inclining towards the Zenith, or upwards 63 Deg. 26 Min. The Center of this Dial is upwards, and the substylar Line is the Hour-Line of 12. The opposite Plane to this, is a North vertical one, inclining downwards or towards the Nadir 63 Deg. 26 Min. and so there is drawn thereon a North inclining Dial, whose Center is downwards.

The Dial C, is a South-East inclining Recliner, whose Declination is 36 Deg. and Inclination to the Zenith 63 Deg. 26 Min. and its Center is downwards. The Dial D is a North-East Decliner of 72 Deg. inclining towards the Nadir 61 Deg. 26 Min. the Center being upwards, and its opposite is a South-West Decliner of 72 Deg. inclining towards the Zenith 63 Deg 26 Min. the Center being downwards.

The Dial E is a North-East Decliner of 36 Deg. and inclines towards the Zenith 63 Deg. 26 Min. the Center being downwards. The opposite Dial to this, is a South-West Decliner of 36 Deg. and inclines towards the Nadir 63 Deg. 26 Min. it’s Center being upwards. Finally, the Dial F is a South-East Decliner of 72 Deg. inclining towards the Zenith 63 Deg. 26 Min. the Center being downwards; and it's opposite is a North-West Decliner of 72 De?. inclining towards the Nadir 63 Deg. 26 Min. the Center thereof being upwards.

All these Dials are furnished with their Axes, which are parallel between themselves, and to the Axis of the World.

Now if one of these Bodies of Dials be set upon a Pedestal, in a Place well exposed to the Sun, and then be set right by means of a Compass or Meridian Line, drawn in the Manner we shall hereafter shew; all the Dials that the Sun shines upon will shew the same Hour or Part at the same Time by the Shadows of the Styles.

But if a Dodecahedron of Dials be to be placed upon a Pedestal fixed in a Garden, it ought to be made of solid Matter, as Stone or good Wood, well painted to preserve it from Rain, &c. therefore it will be here necessary to shew how to cut out a Dodecahedron.

Take a Stone cut out into a perfect Cube, and divide each of the four Sides of it’s Faces into two equal Parts, by two Diameters AC, BD. And at the Points A and C, make the Angles EAF, and HCG, each 116 Deg. 34 Min. that is, make Angles at the Points A and C, on each side the Diameter AC, of 58 Deg. 17 Min each: because all the Surfaces of the Dodecahedron make Angles of 116 Deg. 34 Min. with each other; therefore two Faces thereof being horizontally placed, all the others will incline 63 Deg. 26 Min. the Complement of 116 Deg. 34 Min. to 180 Deg. Now the Space between F and G, or EH, will be the Length of each side of the Pentagons, half of which, viz. BF, must be taken and laid off both ways from the Point I to the Points Q and X. And this must be done upon the Diameters crossing each other on all the other Faces of the Cube. Afterwards the Stone must be cut away along the Diameters to the Extremities of the Sides of the Pentagons: for Example, you must cut away the Stone down, or all along the Diameter KM, in a Right Line to the Point Q in the first Surface of the Cube, as likewise all along the Diameter LN strait forwards to the Point S, and again all along the diameter BD directly forward to the Point T. And proceeding in this manner with the other Faces of the Cube, you may compleat your Dodecahedron. But it will be very proper for a Person that has a mind to cut out one of these Bodies, to have a Pasteboard one before him, thereby to help his Imagination, that so he may know better what Angles and Sides to cut away.

Cylinders may be cut likewise into Dodecahedrons, but let the Method above given suffice.

We make also very curious Dials on the Faces of small brass Dodecahedrons.

The fourth Figure is an Horizontal Dial: To make which, first draw the two Lines AB, CD, cutting each other at Right Angles in the Point E, which will be the Center of the Dial, the Line AB the Meridian or Hour-Line of 12, and the Line CD the Hour-Line of 6. This being done, make the Angle BEF, 49 Deg. equal to the Elevation of the Pole at Paris (the Elevation of the Pole at Paris is but 48 Deg 51 Min. but we neglect the nine Minutes, as being but of small Consequence in the Construction of Dials) and the Line EF will represent the Axis of the World. In this the Point G must be chosen at pleasure, representing the Center of the Earth, and GH must be drawn at Right Angles to EF, cutting the Meridian or Hour-Line of 12 in the Point H. This Line GH represents the Radius of the Equinoctial. Now take H G between your Compass, which lay off from H to B on the Meridian Line, and draw the Right Line LHK perpendicular to the Meridian, which will represent the common Section of the Equinoctial, and the Plane of the Dial: then about the Point B, as a Center, describe the Quadrant MH, which divide into six equal Arcs, each of which will be 15 Deg. and draw the dotted Lines B5, B4, B3, B2, B1. These will divide the Line LH into the Points 1, 2, 3, 4, 5, thro’ which Points, if Lines be drawn from the Center E of the Dial, you will have the Hour-Lines of 1, 2, 3, 4, and 5, on. one side the Meridian; and because the Hour-Lines equally distant on both sides from the Meridian make equal Angles with the Meridian, therefore if the Divisions 1, 2, 3, 4, 5, on one side the Meridian, be laid off from H towards K on the other side, and thro’ the Points where they terminate are drawn Lines from the Center E; these will be the Hour-Lines of 11, 10, 9, 8, 7. And if the Hour-Lines of 7 and 8 in the Morning are continued out beyond the Center, they will give the Hour-Lines of 7 and 8 in the Evening, and likewise the Hour-Lines of 4 and 5 in the Afternoon continued out in the same manner, will give those of 4 and 5 in the Morning. Note, Instead of drawing the Quadrant MH, we might, for greater facility, have only drawn an Arc greater than 60 Deg. for then if an Arc of 60 Deg. had been taken upon it from the Point H, by means of it’s Chord, which is equal to Radius, and the said Arc had been divided in four equal Arcs, each of 15 Deg. and another Arc of 15 Deg. had been added to that of 60 Deg. for the Hour of 5, we might have drawn the Lines B1, B2, B3, &c. as we have already done.

Now to draw the Half-hours, you must bisect each of the Arcs of 15 Deg. on the Quadrant MH, in order to have Arcs of 7 Deg. 30 Min. and for the Quarters, each of these last Arcs must be again bisected; and thro’ each Point of Division occult Lines must be drawn from the Center B, cutting the Equinoctial Line KL. Then if the Edge of a Ruler be laid thro’ these Points of Concourse and the Center E of the Dial, the Halfs and Quarters of Hours may be drawn.

The Hour-Lines being drawn upon your Dial, you may give it what Figure you please, as a Parallelogram, regular Pentagon, &c.

This Dial being fixed upon a very level Plane, that is, set parallel to the Horizon, exposed to the Sun, and it’s Hour-Line of 12 placed exactly North and South; as also the Style or Axis EHF being raised perpendicularly upon the Hour-Line of 12, so as EF be parallel to the Axis of the World: I say, if these things be so ordered, the Shadow of the Axis or Style will shew the Hour of the Day from Sun-rising to Sun-setting.

This Dial is parallel to the Prime Vertical, which cuts the Meridian at Right Angles, and passes thro’ the East and West Points of the Horizon. The Manner of drawing it is thus: First draw the Lines EB and CD at Right Angles, the first of which shall be the Hour-Line of 12, and the other the Hour-Line of 6; then make the Angle BEF at the Point E, the Center of the Dial, equal to the Complement of the Elevation of the Pole, which at Paris is 41 Deg. and raise the Line IG perpendicularly on the Meridian; this will be the right Style, and the Point I is the Foot thereof, and G the Extremity, which, as above said, may be taken for the Center of the Earth: and this Line both ways produced, will be the Horizontal-Line.

From the Point G, in the Right Line EGF, which represents the Axis of the World, raise the Line GH at Right Angles thereto, cutting the Meridian in B. This Line GH shall represent the Radius of the Equinoctial, and the Line LHK, drawn thro’ the Point H, cutting the Meridian at Right Angles, represents the common Section of the Equinoctial and the Plane of the Dial. Now make HB equal to HG, and about the Point B, as a Center, describe the Quadrant of a Circle MH, which divide into 6 equal Arcs, each of which will be 15 Deg. by dotted Lines, dividing the Line LK into unequal Parts, which shall be the Tangents of the said Arcs. Finally, If thro’ those Points of Division and the Center E, you draw Lines, they will be the Hour-Lines on one side of the Meridian; and for drawing the Hour-Lines on the other side the Meridian, as also the Halves and Quarters of Hours, you must do as is shewn in the Horizontal-Dial.

This Dial is set up against a Wall, or on a very upright Plane, directly facing the South; for which reason it is called a Meridional Vertical Dial: it’s Meridional or Hour-Line of 12 must be perfectly upright, and it’s Horizontal-Line level. The Center thereof is upwards, and it’s Axis points towards the under Pole. The opposite Dial to this, is a Vertical North one, having the Center downwards, and the Extremity of it’s Axis pointing to the upper Pole of the World. The Construction of this latter Dial is the same as that of the other, the Flour-Lines and the Axis making the same Angles with the Meridian, as they do on that. But the Sun shines but a small time upon this Dial, and this only in the Summer-time, viz. in the Morning from his rising ’till he has passed the Prime Vertical, and in the Evening from the time he has again passed the Prime Vertical ’till his setting. When the Sun describes the Summer Tropick, he rises at Paris, at 4 in the Morning, and comes to the Prime Vertical between 7 and 8 in the Morning; and in the Afternoon he repasses the Prime Vertical between 4 and 5, and sets at 8. Therefore we need only draw the Hour-Lines upon this Dial from 4 in the Morning to 8, and from 4 in the Afternoon to 8; at which time the Sun shines upon the Meridional Vertical Dial, but from about 8 in the Morning to about 4 in the Afternoon. But when the Sun by his annual Motion is again come back to the Equinoctial, he will not shine at all upon the Vertical North Dial till after he has crossed the Equinoctial again; and all this time he will shine upon the Meridional Vertical Dial from his rising to his setting.

The 6th Figure represents an upper Polar Dial, which is one that inclines upwards, but does not decline: for it is parallel to the Axis of the World, and the Hour-Circle of 6, which cuts the Meridian at Right Angles. And for this reason the Hour of 6 in the Morning or Evening can never be shewn by this Dial; for the Shadow of the Style being then parallel to the Plane of the Dial, cannot be cast upon it. This Dial likewise hath no Center, and the Hour-Lines are all parallel between themselves, and to the Axis of the World. The Plane therefore being parallel to the Horizon of a right Sphere, passes thro’ the two Poles of the World, from whence comes the Name of a Polar Dial.

The Manner of drawing this Dial is thus: First draw the Line AB representing the Equinoctial, and ID at Right Angles thereto, for the Meridian or Hour-Line of 12. Then assume the Length of the Style at pleasure, according to the bigness of the Plane the Dial is to be drawn on; let this be CD, about the Extremity of which D describe a Quadrant, which divide into six equal Arcs (or only describe an Arc of 60 Degrees, which divide into four Parts, of 15 Degrees each, for the four first Hours after Noon, and then add an Arc of 15 Degrees for the flour of 5.). This being done, draw dotted Lines from the Point D, thro’ the Divisions of the Circumference of the said Arc, to the Line AB; and then if Lines are drawn thro’ the Points wherein the dotted Lines cut the Line AB, parallel to the Meridian, these Lines will be the Hour-Lines on one side the Meridian: and if there be as many Parallels drawn on the other side the Meridian, at the same Distances therefrom as the respective parallel Hour-Lines are on the other side, these will be the Hour-Lines on the other side of the Meridian. The Style of this Dial must be equal in Length to CF, the Distance from the Hour-Line of 3 to the Hour-Line of 12, and may be made in figure of a Right-angled Parallelogram, as is that marked above the Letter K in the Figure of the Dial. This Style is let upon the Hour-Line of 12, which for this Reason is called the Substylar Line.

If a single Rod only be used for a Style, as that which is in the Point C of the Meridian, then the Hour will be shewn upon this Dial by the Shadow of the Extremity of the Style; whereas when a Parallelogram is used, we have the Hour shewn by the Shadow of one of its Sides, that is, by a right Line.

An upper Polar Dial may shew the Hour from seven in the Morning to five in the Afternoon; and an under Polar one is useless, unless in the Summer, wherein the Hour is shewn thereby, from the Sun’s rising to five in the Morning, and from seven in the Evening ’till his setting: and so for the Elevation of the Pole of Paris, the Hours of four and five in the Morning, and seven and eight in the. Afternoon, are only set down upon this Dial; and these may be drawn as those on the upper Polar Dial, for the Distances of the Hour-Lines of four and five in the Afternoon from the Substyle, on the upper Polar Dial, are equal to the Distances of the Hour-Lines of four and five in the Morning from the Substyle on the under Polar Dial. Understand the same for the Hours of seven and eight in the Afternoon; and therefore there is no need of drawing the Figure of this Dial. Note, The Distance of the Hour-Lines on these Dials depend upon the Breadth of the Style, or the Distance of the Point D from the Equinoctial Line.

To set up this Dial at Paris, the Plane thereof must make an Angle of 49 Deg. with the Horizon, the upper one facing the Sky directly South, that so the Axis thereof may be parallel to that of the World, and the opposite Dial to this, viz. the under Polar one faces downwards, the Morning Hours being, towards the West, and the Afternoon ones towards the East, on both the upper and under ones.

Now if the Horizontal Line is to be drawn upon this Dial, describe the Arc GH, about the Point F, the Extremity of the Style, equal to the Elevation of the Pole, viz. 49 Deg. for the Latitude of Paris, and draw the Right Line FH, cutting the Meridian in the Point I, thro’ which draw the Horizontal Line LK, at Right Angles. Now by means of this Line, we may know whether the Dial be well placed, and have its convenient Inclination; for if the Dial be inclined rightly, a Plane laid along the Horizontal Line, and supported by the Edge of the Style, will be level or parallel to the Horizon.

A Polar Dial in a right Sphere is parallel to the Horizon, and in a parallel Sphere it is vertical or upright.

An upper Equinoctial Dial shews the Hour but only six Months in the Year, viz. from the Vernal Equinox to the Autumnal one; and the opposite Dial to this, which is an under Equinoctial one, shews the Hour during the other six Months of the Year, viz. from the Autumnal Equinox to the Vernal one.

The Plane of this Dial is parallel to the Equinoctial Circle, and is cut at Right Angles through the Center thereof by the Axis of the World.

The Construction of this Dial is thus: Draw two Right Lines AH, and ED, crossing each other at Right Angles, the first of which shall be the Hour-Line of 12, and the other the Hour-Line of 6; then about the Point A of Intersection describe a Circle, each quarter of which divide into six equal Parts, thro’ which, if strait Lines be drawn from the Center A, these Lines will be the Hour-Lines, because they each make equal Angles of 15 Deg. and if each of these Angles be halved and quartered, the halves and quarters of Hours will be had.

The Construction of an under Equinoctial Dial is the same as of an upper one; and in a parallel Sphere, viz. where the Pole is in the Zenith, there is but one Equinoctial Dial, which will likewise be there an Horizontal one. And in a right Sphere, viz. where the two Poles are in the Horizon, these Dials are non-declining Vertical ones, are set up against Walls, one of which faces the North Pole, and the other the South Pole, the Sun shining upon each six Months in the Year. But in an oblique Sphere, as that which we inhabit, these Dials are inclined to the Horizon, and make an Angle therewith equal to the Complement of the Latitude, viz. at Paris, an Angle of 41 Deg.

The Axis of an Equinoctial Dial is a strait Iron Rod going thro’ the Center of the Dial perpendicular to the Plane thereof, and parallel to the Axis of the World. The Length of this Rod may be at pleasure, when it hath no other Use but shewing the Hour by the Shadow thereof; but when the Length of the Days, and the Sun’s Place are to be shewn thereby, the said Rod must have a determinate Length, as we shall shew hereafter.

These Dials are parallel to the Plane of the Meridian; one of which directly faces the East, and the other the West. The 8th Figure is a West Dial, having the Hour-Lines parallel to each other, and to the Axis of the World, as in a Polar Dial, and their Construction is nearly the same as of the Hour-Lines on a Polar Dial.

This Dial is made thus: First draw the right Line AB, representing the Horizontal Line, and about the Point A, assume the Arc BC of a Radius at pleasure in this Line, equal to the Complement of the Latitude, or Height of the Equator above the Horizon, which at Paris is 41 Deg. Then draw the Line CD, produced, as is necessary, from the Point C, and this Line shall represent the common Section of the Equinoctial and Plane of the Dial; after this, draw ED from the Point D, parallel to the Equinoctial Line, and this Line ED will be the Place of the Substyle, that is, the Line on which the Style must be placed; as likewise the Hour-Line of Six. Now to draw the other Hour-Lines, assume the Point E at pleasure on the substylar Line, about which, as a Center, describe an Arc of 60 Deg. which divide into four equal Parts for 15 Deg. each, beginning from the substylar Line. After this, lay off as many Arcs of 15 Deg. as is necessary upon the said Arc both ways continued, and draw dotted Lines from the Center E thro’ all the Divisions of the Arc to the Equinoctial Line: then if right Lines be drawn thro’ the Points in the Equinoctial Line, made by the dotted Lines, parallel to the Hour-Line of 6, and perpendicular to the Equinoctial Line; these Lines will be the Hour-Lines. Note, This Dial shews the Time of Day after Noon to the Setting of the Sun; and since the Sun sets (at Paris) at Eight o’Clock in the Summer, we have pricked down the Flour-Lines from One to Eight in this Dial, as appears per Figure.

The Construction of an East Dial is the same as of this; and there are pricked down the Hour-Lines upon it from the Sun’s rising in Summer, viz. from Four in the Morning to Eleven. The reason that the Hour-Line of Twelve cannot be drawn upon these Dials, is, because when the Sun is in the Meridian, his Rays are parallel to their Planes.

If a West Dial be drawn upon a Sheet of Paper, and then the said Paper is rendered Transparent by oiling, you will perceive thro’ the backside of the Paper an East Dial drawn entirely; only the Figures of the Hours must be altered, that is, you must put 11 in the Place of 1; 10 in the Place of 2; and so of others.

The Style of these Dials is a Brass or Iron Rod, in Length equal to ED, which is likewise equal to the Distance of the Hour of 3 from the Hour of 6. This Style is set upright in the Point D, and shews the Hour by the Shadow of it’s Extremity. These Dials, which may have likewise a Style in figure of a Parallelogram, as we have mentioned in speaking of Polar Dials, are set upright against Walls or Planes, perpendicular to the Horizon, and parallel to the Meridian, one of which directly faces the East, and the other the West, in such manner, that the Horizontal Line be perfectly level.

Now suppose we know that a Plane (as that marked I of Figure 1.) or upright Wall, declines 45 Deg. South-westwardly at Paris, or thereabouts, where the Pole is elevated 49 Deg. above the Horizon. It is required to draw a Dial for this Declination.

First, draw the Lines AB, CD, crossing each other at Right Angles in the Point E, the former of which shall be the Hour-Line of 12, and the other the Horizontal Line. About the Point E, as a Center, draw the Arc FN of 45 Deg. because the Plane’s Declination is such, and since it is South-westwardly, the said Arc must be drawn on the Right-side of the Meridian; but if the Declination had been South-eastwardly, that Arc must have been drawn on the Left-side the Meridian. This being done, raise the Perpendicular FH from the Point F to the horizontal Line, that so we may have one Point of the Style therein, viz. the Foot of the Style. Then take the Distance EF between your Compasses, and lay it off upon the horizontal Line from E to O, and about the Point O, as a Center, describe the Arc EG equal to the Height of the Pole, viz. in this Case 49 Deg. and draw the dotted Line OA to the Hour-Line of 12; then A will be the Center of the Dial thro’ which the Substyle AH must be drawn of an indeterminate Length. Note, This Substyle is one of the principal Lines, by means of which a Dial of this kind is drawn, and upon which the whole Exactness thereof almost depends.

Upon the Point H raise the right Line HI equal to HF, perpendicular to the Substyle AH, and draw the right Line AI, prolonged, for the Axis of the Dial. Then let fall the Perpendicular KI to the Axis, cutting the substylar Line in K, and make KL equal to KI, and draw a right Line both ways thro’ the Point K, perpendicular to the Substyle AK; this will represent the Equinoctial Line, and cuts the Horizontal Line in a Point thro’ which the Hour-Line of 6 must pass. Thus having already the Hour-Lines of 12 and 6, if the Operations hitherto performed have been done right, two dotted Lines L6, and LN being drawn, will be at Right Angles to each other. Again, About the said Point L, as a Center, describe the Quadrant of a Circle between the said dotted Lines, whole Circumference divide into 6 equal Arcs, of 15 Degrees each, and draw occult Lines thro’ the Points of Division to cut the Equinoctial Line; but to have the Morning Hour-Lines, and those after 6, prolong the Arc of the Quadrant both ways, and lay off as many Arcs of 15 Degrees upon it, as is necessary, that so occult Lines may be drawn from the Center L to cut the Equinoctial Line. Then if Lines are drawn from the Center A thro’ all the Points wherein. the occult Lines cut the Equinoctial Line, these Lines thus drawn will be the Hour-Lines. Note, There must be but 12 Hour-Lines drawn upon any vertical declining Plane, for the Sun will shine on any one of them but 12 Hours.

Points in the horizontal Line DC, thro’ which the Hour-Lines must pass, may be found otherwise, by applying the Center of a horizontal Dial to the Point F, in such manner, that the Meridian Line thereof coincides with the Line FE, and it's Hour-Line of 6, with the Line F6: for then the Points where the Hour-Lines of the horizontal Dial cut the said Line DC, will be the Points therein thro’ which the Hour-Lines must be drawn from the Center A.

The Hour-Lines of six Hours successively being given upon the Plane of any Dial what-soever, the other Hour-Lines may be drawn after the following Manner: Suppose, in this Example, that the Hour-Lines from 6 to 12 are drawn; now if you have a mind to draw the Hour-Lines of 9, 10 and 11 in the Morning, which may be pricked down upon this Dial, draw a Parallel, as SV, from the Point V, taken at pleasure on the Hour-Line of 12, to the Hour-Line of 6, which shall cut the Hour-Lines of 1, 2, and 3, in the Afternoon. This being done, the Distance from V to the Hour-Line of 1 taken on this Parallel, and laid off on the other Side, will give a Point in the said Parallel thro’ which the Hour-Line of 11 must be drawn; likewise the Distance V2 will give a Point thereon, thro’ which the Hour Line of 10 must be drawn; and the Distance V3 will give a Point thro’ which the Hour-Line of 9 must pass. And so if Lines are drawn from the Center of the Dial A thro’ the said Points, they will be the Hour-Lines.

In this manner likewise may be found the Points thro’ which the Hour-Lines of 7 and 8 in the Evening are drawn, in first drawing a Parallel to the Hour-Line of 12, cutting the Hour-Line of 6 in one Point, and meeting the Hour-Lines of 4 and 5 produced; for the Distance from the Points where the Hour-Lines of 6 and 5 are cut by this Parallel, laid off on the other Side from the Point where the Hour-Line of 6 cuts the Parallel, will give a Point upon it thro’ which the Hour-Line of 7 must be drawn. And the Distance from the Points where the Parallel cuts the Hour-Lines of 6 and 8, laid off on the other Side on that Parallel, will give a Point therein thro’ which the Hour-Line of 8 must pass; and if Lines are drawn from the Center A thro’ those two Points found, they will be the Hour-Lines of 7 and 8 in the Evening. This is a very good way of drawing those Hour-Lines that are pretty distant from the substylar Line, because thereby we avoid cutting the Equinoctial very obliquely.

The Construction of a South-East vertical Decliner is the same as of that which we have described, excepting only that what was there made on the Right must here be on the Left, and the Figures for the Morning Hours set to those for the Afternoon: so that if a South-West declining Dial be drawn upon a Sheet of Paper, and afterwards the Paper be oiled, that you may see thro’ it, you will see a South-East Decliner thro’ the Paper; only the Figures set to the Hour-Lines must be altered; as, where the Figure of 1 stands, you must set 11; where the Figure of 2, 10; where the Figure of 3, 9; and so on. By this means the substylar Line, which falls between the Hour-Lines of 3 and 4 Afternoon, in Figure 9, will fall in this Dial between 8 and 9 in the Morning. And if the Plane’s Declination had been less than 45 Deg. the Substyle would have fallen yet nearer to the Meridian: but if, on the contrary, the Declination thereof had been greater, the Substyle would have fallen more distant from the Meridian, and pretty near the Hour-Line of 6. But when this happens, the Hour-Lines fall so close together near the Substyle, that we are obliged to make the Model of a Dial upon a very large Plane, that so the Hour-Lines may be very long, and the part of the Dial towards the Center taken away.

After the abovenamed manner, likewise may be drawn North-East and North-West Dials; but these have their Centers downwards underneath the Horizontal Line, and properly are no other but South-East or South-West Decliners inverted, as may be seen in Figure 10, which represents a North-West Decliner of 45 Deg. drawn for the Plane L of Figure 1. and the substylar Line of this Dial must be between the Hours of 8 and 9 in the Evening, whence one Decliner only may serve for drawing four, if they have an equal Declination, tho’ to different Coasts; two of which will have their Centers upwards, and the other two their Centers downwards.

Suppose the Substylar Line is to be found on the Decliner of Figure 9, first place obliquely upon the Dial-Plane, a Wire or Iron Rod, sharp at the end, so that the Extremity thereof be perpendicularly over the Point H in the Plane. This may be done by means of a Square.

Now since this Figure is a South-West vertical Decliner, therefore the Substylar Line thereon must be found among the Afternoon Hours, to the Right-hand of the Meridian; and consequently, let us suppose the Shadow of the Extremity of the Iron-Rod at the first Observation to fall on the Point P; then about the Point H, the Foot of the Style, with the Distance HP, describe the circular Arc PR. This being done, some Hours after the first Observation the same Day, observe when the Shadow of the Extremity of the Rod falls a second time upon the aforesaid Arc, which suppose in the Point Q: then if the Arc PQ be bisected in the Point R, and a Right-line be drawn thro’ the Points R and H; this Line will be the Substyle, which being exactly drawn, and the Height of the Pole above the Horizon of the Place where the Dial is made for, being otherwise known, it will not then be difficult to compleat the Dial; for first, the Meridian or Hour-Line of 12 is always perpendicular to the Horizon, in vertical Planes, and the Point wherein the Meridian and Substylar Line produced meet each other, (as the Point A) will be the Center of the Dial. The Horizontal Line is a level Line passing thro’ the Foot of the Style, as DHC.

And to draw the Equinoctial Line, you must first form the Triangular Style AHI on the Substyle, whose Hypothenuse AI is the Axis, and Side HI the right Style; then if IK be drawn from the Point I perpendicular to the Axis, meeting the Substylar Line in the Point K; and if thro’ K a Right Line MKN be drawn at Right Angles to the Stylar Line, this Line will be the Equinoctial, and the Point wherein it cuts the Horizontal Line will be always the Point thro’ which the Hour-Line of 6 must pass. Moreover, the Distance KL, laid off on the Stylar Line, will give the Point L the Center of the Equinoctial Circle. Now what remains to be done, may be compleated as before explained; and even the whole Dial may be drawn in one’s Room, after the Positions and Concourses of the principal Lines are laid off upon a Sheet of Paper, and the Angle which the Substylar Line makes with the Meridian or Horizontal Line be taken; for one is the Complement of the other.

Now to prove whether the Equinoctial Line be drawn right, make the Angle BAO equal to the Complement of the Elevation of the Pole, viz. 41 Deg. for the Latitude of Paris, draw the Line AO to the Horizon, and make the Angle AON a Right one, that so the Point N may be had in the Meridian or Hour-Line of 12, thro’ which the Equinoctial Line must pass. Thus having several Ways for finding the principal Points, one of them will serve to prove the other.

When a Dial Plane declines South-eastwardly, the Substylar Line will be on the right Side of the Meridian. In which Case it is proper to take notice, that in finding the Substylar Line, as above, to observe when the Shadow of the Extremity of the Rod falls upon the Plane, as soon as the Sun begins to shine thereon; as likewise to mind the Time very exactly when the Shadow of the Extremity of the Style comes again to touch the circular Arc; you may operate in this manner several Days successively, in order to see whether the Position of the Substylar Line has been found exactly.

When a Plane declines North-East or North-West, the Shadows of the Extremity of the Iron Rod fall above the Foot of the Style, and so the Center of the Dial must be down-wards. Likewise the most proper Time for making these Operations is about 15 Days before or after the Solstices, for when the Sun is near the Equinoctial, his Declination is too sensible, and the Operations less exact. Nevertheless the Equinoctial Line may be drawn upon a Plane, when the Sun is in the Equinoctial Points, and by that means a vertical declining Dial constructed, by the following Method.

In order for this, let it be required to draw a Dial upon a Plane of the abovesaid Declination and Inclination. First, Draw the two Lines AB, CD, cutting each other at right Angles in the Point E; then let CD be parallel to the Horizon, and upon it assume EF at pleasure, for the Length of the right Style, whose Foot shall be E, and Extremity F, and about the Center F describe the Arc GH, equal to the Plane’s Inclination, viz. 63 Deg. 26 Min. and draw the right Line AF; likewise make the Angle GFI equal to the Complement of 63 Deg. 26 Min. viz. 26 Deg. 34 Min. This being done, the Point A will be the Nadir, and one Point of the Meridian Line, and if a right Line MLN be drawn thro’ the Point L, parallel to CD, this will be the horizontal Line; and if the Distance LF be taken between your Compasses, and laid off from L to O, the Point O will be the Center thro’ which Lines may be drawn dividing the horizontal Line. Again, About the Point O describe the Arc LP of 36 Deg. viz. the Plane’s Declination, and draw the Line OP cutting the horizontal Line MLN in the Point 12; then if a right Line be drawn thro’ the Nadir A and this Point 12, the said Line A12 will be the Meridian of the Dial or Hour-Line of 12: and moreover, if an Angle be made at the Point O on the Left-side of the Line AB, equal to the Complement of the Plane’s Declination, which here is 54 Deg. you will have a Point on the horizontal Line thro’ which the Hour-Line of 6, as likewise the Equinoctial Line, must pass.

The next thing to be found is another Point, besides E the Foot of the Style, thro’ which the substylar Line must pass; and in order for this, we need only find the Center of the Dial, after the following manner.

Draw the Line MR from the Point M, (thro’ which the Hour-Line of 6 passes) at right Angles to the Meridian A12, lay off the Distance O12, from 12 to R, or else the Distance AF from A to R, draw the occult Line 12R, and about the Point R describe the Arc NK, of 49 Deg. viz. the Elevation of the Pole; then if RK be drawn cutting the Meridian in the Point K, this will be the Center of the Dial. After this, the Substylar Line KE may be drawn; and if the Perpendicular MQ be drawn to this Line thro’ the Point M, the said MQ will be the Equinoctial Line. Moreover, the Point in the Meridian Line thro’ which the Equinoctial Line must pass, may be found by making the Angle NRQ of 41 Deg. that is, the Complement of the Elevation of the Pole.

The Positions of the principal Lines being thus found, it will not now be difficult to find the Points on the horizontal or equinoctial Lines, thro’ which the Hour-Lines must be drawn; for if the Points are to be found upon the horizontal Line you must apply the Center of a horizontal Dial to the Point O, in such manner, that the Hour-Line of 12 answers to the Line O12, and the Hour-Line of 6 to the Line O6: then the Points in the horizontal Line MN, thro’ which the other Hour-Lines must be drawn, may be determined easily. And if the Points thro’ which the Hour-Lines must pass on the equinoctial Line be to be found, you must raise the Perpendicular ES on the Substyle equal to EF, and draw the Axis SK; and afterwards take the Distance TS between your Compasses, and lay off on the Substyle from T to V, then V will be the Center of the Equinoctial Circle, by means of which the Equinoctial Line may be divided, as we have directed in speaking of declining Dials, and the Hour-Lines drawn thro’ the Center of the Dial K.