Mathematical Instruments
Book VIII. Ch. I.

# 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.

## The Construction of Dials drawn upon a Dodecahedron.

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 Construction of an Horizontal Dial.

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.

## The Construction of a Non-declining Vertical Dial.

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 Construction of a Non-declining Vertical Dial.

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.

## The Construction of an Equinoctial Dial.

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.

## The Construction of East and West Dials.

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.

## The Construction of Vertical Declining Dials.

A Vertical Dial is one that is made upon a Vertical Plane, that is, a Plane perpendicular to the Horizon, as a very upright Wall.

Among the nine Regular Dials of which we have spoken, there are four of them Vertical ones, which do not decline at all, since they directly face the four Cardinal Parts of the World. It now remains that we here speak of Irregular Dials, some of which are vertical Decliners, others undeclining Decliners, and finally, others declining Incliners. Vertical Decliners are of four Kinds: for some decline South-eastwardly, the opposite ones to these, North-westwardly, others decline South-westwardly, and the opposite ones to these, North-eastwardly.

Now among the Irregular Dials, the vertical Decliners are most in use, because they are made upon or set up against Walls (which commonly are built upright), or else upon Bodies whose Planes are upright; but before these Dials can be made, the Declinations of the Walls or Planes, on which they are to be made or set up against, must first be known or found exactly: and this may be done by some one of the Methods hereafter mentioned.

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.

## To draw the Substylar Line upon a Plane by means of the Shadow of the Extremity of an Iron-Rod, observed twice the same day.

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.

## To draw the Equinoctial Line upon a vertical Plane by means of the Shadow of the Extremity of an Iron-Rod.

The most simple and easy Method to draw the Equinoctial Line upon a Wall or Plane, is at the Time when the Sun is in the Equinoctial, (though this may be done at any other Time by more complicated Methods) for when the Sun describes the Equinoctial by his diurnal Motion, the Shadows of the Extremity of the Iron-Rod or Style, will all fall upon a Plane in a right Line, which is the common Section of the Equinoctial Circle, of the Heavens and the Plane. Therefore if several Points, pricked down upon a Plane, made by the Shadow of the Extremity of the Rod, on the Day the Sun is in the Equator, be joined, the right Line joining them will be the Equinoctial Line, as the Line MN, in Figure 9. This being done, draw the right Line AHL thro’ the Foot of the Style at Right Angles to the Equinoctial Line, and this will be the Substylar Line: Moreover, draw the level Line DHC thro’ the Foot H of the Style; this will be the Horizontal Line; and if HI be drawn equal to the Height of the right Style, and parallel to the Equinoctial Line and the Points K and L joined; and if AI be drawn at Right Angles to KI, then the Point A will be the Center of the Dial, and the upright Line AB the Meridian or Hour-Line of 12. The common Section of the Equinoctial and Horizontal Lines, will likewise be the Point thro’ which the Hour-Line of 6 must pass, and consequently wherewith the Dial may be finished. Note, The Angle HFE will be the Plane’s Declination.

## To draw a Dial upon a Vertical Plane by means of the Shadow of the Extremity of an Iron-Rod or Style observed upon the Plane at Noon.

A Style, as HI (Vide Figure 9.), being set up on a Wall or Dial Plane, whose Foot is H, and Extremity I; and if you know by any means when it is Noon, which may be known by a Meridian Line drawn upon a Horizontal Plane, as we shall mention hereafter, note where the Extremity of the Shadow of the Style HI falls upon the Plane at Noon, which suppose in the Point N, and thro’ this Point draw the Perpendicular ANB, which consequently will be the Meridian of the Place or Hour-Line of 12; then draw the level Line CHD, cutting the Meridian at Right Angles in the Point E; this will be the Horizontal Line. Again, Draw HF equal in Length to the right Style HI, and parallel to the Meridian; then take the Hypothenuse EF between your Compasses, and lay it off upon the Horizontal Line from E to O, and make the Angle EOA equal to the Elevation of the Pole, viz. 49 Deg. and then the Point A will be the Center of the Dial.

Likewise make the Angle EON, underneath the Horizontal Line, equal to the Complement of the Elevation of the Pole, viz. 41 Deg. and the Point N on the Meridian Line will be that thro’ which the Equinoctial Line must pass. Then if the right Line AHK be drawn thro’ the Center A, and the Foot of the Style H, this will be the Substylar Line; and if a Perpendicular be drawn thro’ the Point N to this Line, the said Perpendicular will be the Equinoctial Line. Thus having found the principal Lines of the Dial, you may compleat it by the Methods before explained.

This Method of drawing a Dial at any Time of the Year, by means of the Shadow of the Extremity of the Style HI observed at Noon, may serve, when it is not possible to find the Substylar Line by the Observations of the Shadows of the Extremity of an Iron-Rod or Style, which happens when Planes decline considerably Eastwards or Westwards.

There are several other Methods of drawing Vertical Dials on Walls or Planes: but those would take up too much time to mention in this small Treatise, wherein we have only laid down the most simple and easy Methods of drawing Vertical Dials. And in order to draw Dials more exactly, we shall hereafter lay down Rules for calculating the Angles the Hour-Lines make at the Centers; and so the other Methods may be verified by these Rules.

## The Construction of Non-declining inclining Dials.

The Inclinations of these Dials are the Angles that their Planes make with the Horizon, and some of them face the Heavens, and others the Earth. There are likewise two Kinds of them with regard to the Pole; and two other Kinds with regard to the Equinoctial.

If a Plane facing the South hath an Inclination towards the North, this Inclination may be less or greater than the Elevation of the Pole; for if the Inclination be equal to the Elevation of the Pole, this Dial-Plane will be an upper or under Polar one, whose Construction we have already laid down.

If the Inclination be less than the Elevation of the Pole, which at Paris is nearly 49 Deg. and you would make a Dial upon a Plane facing the South, having 30 Deg. of Inclination towards the North, substract 30 Deg. from 49 Deg. and the Remainder 19 Deg. will be the Height of the Axis or Style above the Plane. Then if a Horizontal Dial be made upon this Plane for the Latitude of 19 Deg. in the manner we have already laid down, we shall have an Incliner of 30 Deg. drawn, because the said Plane thus inclined is parallel to the Horizon of those Places where the Pole is elevated 19 Deg. and consequently this must be a Horizontal Dial for those Places. The Center of this Dial is downwards, underneath the Equinoctial Line, and the Morning Hour-Lines on the Left, and the Afternoon ones on the Right-hand of those looking at them.

The under opposite Dial to this, which faces towards the North, is the same as the upper one facing towards the South, excepting only that the Center is upwards above the Equinoctial Line, and the Morning Hour-Lines on the Right, and the Afternoon ones on the Left-hand.

If the Inclination of the Plane be greater than the Elevation of the Pole, suppose at Paris, and it be 63 Deg. substract the Elevation of the Pole 49 Deg. from 63 Deg. and the Remainder will be 14 Deg. and then make an Horizontal Dial for this Elevation of 14 Deg. and you will have an Incliner of 63 Deg. the Center of the upper Plane facing towards the South, is upwards above the Equinoctial Line, the Morning Hour-Lines on the Left-hand, those of the Afternoon towards the Right; and in the opposite under Plane facing towards the North, the Center is downwards, the Morning Hours on the Right, and those of the Afternoon on the Left, as may be seen in Figure 11 and 12.

If the Plane faces the North, and inclines Southwards, the Inclination thereof may be less or greater than that of the Equinoctial; for if it be equal, we need only make an upper or under Equinoctial Dial thereon, which is a Circle divided into 24 equal Parts, as is above directed in speaking of Regular Dials.

If the Inclination be less than the Elevation of the Equinoctial, as, suppose a Plane at Paris inclines 30 Deg. Southwardly, add the 30 Deg. of Inclination to 49 Deg. the Height of the Pole, and make an Horizontal Dial for the Elevation of 79 Deg. and your Dial will be drawn: the Center of the upper Dial facing Northwardly, will be upwards, the Morning Hour-Lines on the Right-hand, the Afternoon ones on the Left; and on the opposite under Dial to this, the Center will be downwards, the Morning Hour-Lines on the Left, and the Afternoon ones on the Right-hand.

Finally, If the Inclination, which suppose 60 Deg. be greater than the Height of the Equinoctial, add the Complement of the Inclination, which is 30 Deg. to the Elevation of the Equinoctial, which is 41 Deg. at Paris, and the Sum is 71 Deg. and make an Horizontal Dial for this Elevation of the Pole. The Center of the upper one of these Dials is downwards, the Morning Hour-Lines on the Right-hand, and the Center of the opposite under Dial is upwards, and the Morning Hour-Lines on the Left-hand.

Note, The Meridian or Hour-Line of 12, is the Substylar Line of all Non-declining inclining Dials, passes thro’ their Centers at right Angles to the Hour-Lines of 6, and may be drawn by means of the Shadow of a Plumb-Line passing thro’ their Centers.

There ought to have been eight Figures to represent all these different Dials, viz. four for the upper ones, and four for the under ones; but since they are not difficult to be conceived or drawn, we have only represented two of them, with respect to the Dodecahedron on which we place them.

## The Construction of declining inclining Dials.

The Declination of a Dial is the Angle that the Plane thereof makes with the Prime Vertical; and it’s Inclination is the Angle made by the Plane thereof with the Horizon: both of which we shall shew how to find hereafter.

Now suppose, for Example, that a Dial is to be drawn upon a Plane declining 36 Deg. South-eastwardly, and inclining 63 Deg. 26 Min. towards the Earth, as does the Plane C on the Dodecahedron of Figure 2.

But before we shew how to draw this Dial, you must first observe that the Horizontal Line, which passes thro’ the Foot of the Style in Vertical Dials, must in no wife pass thro’ it in inclining Dials; for in upper Incliners facing the Heavens, this Line must be drawn above the Foot of the Style, and in under Incliners, facing the Earth, below the Foot of the Style. Secondly, The Meridian or Hour-Line of 12, in inclining Dials, does not cut the Horizontal Line at right Angles, as it does in Vertical Dials, but must be drawn thro’ two Points; one of which is found upon the Horizontal Line by means of the Angle of Declination, and the other upon a Vertical Line cutting the Horizontal one at right Angles.

This last Point in upper Incliners is called the Zenith Point, because if the Sun was in the Zenith of the Place for which the Dial is made, the Extremity of the Shadow of the Style would fall upon that Point, which consequently will be underneath the Style of these Dials. And in under Incliners, the said Point is called the Nadir Point, because if the Sun was in the Nadir, and the Earth transparent, the Extremity of the Shadow of the Style would touch that Point, which consequently will be above the Style, as in the proposed Dial.

Thirdly, The Center of the proposed under Dial which declines South-eastwardly must be upwards, the Substylar Line to the Left-hand of the Vertical Line, and the Meridian among the Morning Hour-Lines, and so on the Right of the Vertical Line. The Centers of upper Dials declining South-westwardly must be likewise upwards, the Substylar Line on the Right-hand of the Vertical one, and the Meridian among the Afternoon Hour-Lines; and the opposite upper Dials to these, have their Centers downwards, and are no other but these Dials inverted: and therefore one of these four Dials is enough to be drawn.

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.

Your Dial being thus made, you may draw a fair Draught thereof, wherein are only the principal Lines, and the Hour-Lines, as may be seen in the Pentagonal Figure marked 14.

The Dial of the Figure 15, represents that marked F in Figure 2, and is an upper Incliner of 63 Deg. 26 Min. declining South-eastwardly 72 Deg. and may be drawn by the abovesaid Method. The Center of this Dial is upwards, and because it has a great Declination, the Hour-Lines will fall very close to one another near the Substylar Line; and therefore it ought to be drawn upon a large Plane, that so the Part thereof next to the Center may be taken away, and the Style and the Hour-Lines terminated by two Parallels.

There is another way of drawing Mechanically any sorts of Dials whatsoever, upon Polyhedrons or Bodies of different Faces or Superficies, without even knowing the Declinations or Inclinations of the Faces or Superficies, and that with as much exactness as by any other Methods whatsoever. In order to do this, you must first make an horizontal Dial upon one of the Planes or Faces that is to be set parallel to the Horizon, and set up the Style thereof upon the Hour-Line of 12, conformable to the Latitude of the Place. After this, the Substylar Lines must be drawn upon all the Planes or Faces of the Polyhedron that the Sun can shine upon, that so Brass or Iron Styles, proportioned to the bignesses of the Planes, or Faces, may be fixed upon them perpendicularly in such manner, that the Axes or upper Edges of the said Styles be parallel to the Axis of the horizontal Dial. This may be done in filing them away in right Lines by Degrees, until their Axes, being compared with the Axis of a large Style similar to that of the horizontal Dial placed level, (or held up so that its Base be parallel to the Horizon, by means of a Thread and Plummet hung to the Top of the Style) appear in a right Line with the Axis of the said Style.

Things being thus ordered, set your Polyhedron in the Sun, and turn it about, making the Shadow of the Axis of the horizontal Dial fall upon each Hour-Line thereof successively, and if at each of the respective Times right Lines be drawn along the Shadows of the Axes of the Styles of the other Faces of the Body upon the said Faces, these will be the same Hour-Lines upon each of the Faces of the Body, that the Shadow of the Style of the Horizontal Dial fell upon, on the Horizontal Dial. For example; Suppose the Shadow of the Axis of the Horizontal Dial falls upon the Hour Line of 12; then at the same time draw Lines along the Shadows of the Styles upon the other Faces of the Body, and those Lines will be the Hour-Lines of 12 upon the said Faces: understand the same for others. This may be done likewise in the Night, by the Light of a Link moved about the Polyhedron.

There are great Stone Bodies cut into several Faces placed sometimes in Gardens having Dials drawn upon them, according to the abovesaid Method, And the Edges of the Stone which serve for Axes to some of these Dials, must be cut so as to be parallel to the Axis of the World.

## The Arithmetical Construction of Dials by the Calculation of Angles.

This Method is a great help for verifying any Operations in Dialling, wherein there is great Exactness required, and chiefly when we are obliged to make a small Model for drawing a large Dial: for an Error almost insensible in the Model, will become very considerable in the long Hour-Lines to be drawn upon a large Plane.

In the Construction of Regular Dials, as of the Horizontal one of Figure 4, the Divisions of the Equinoctial Line LK, are the Tangents of the Angles of the Quadrant MH, and the dotted Lines are their Secants; and therefore they may be pricked down by means of a Scale or Sector, in supposing the Radius HB100: for then the Tangent H1 of 15 Deg. will be twenty-seven of the said Parts; H2, the Tangent of 30 Deg. will be 58; H3, the Tangent of 45 Deg. (equal to Radius) will be 100; H4, the Tangent of 60 Deg. will be 173; and H5, the Tangent of 75 Deg. will be 373 Parts. The Divisions on the other half of this Line for the Morning Hour-Lines are the same.

The Divisions for the halves and quarters of Hours may be found likewise upon the Equinoctial Line, by assuming the Tangents of the correspondent Arcs, which may be taken from printed Tables of natural Tangents, but from the Table of Secants we can deduce some Abreviations. For example, the Line B4, which is the Secant of 60 Deg. being double to Radius, if twice BH be laid off from B4, you will have the Point on the Equinoctial Line thro’ which the Hour-Line of 4 must be drawn. The said Secant laid off from 4 to L, will give likewise the Point in the Equinoctial Line thro’ which the Hour-Line of 5 must be drawn, &c.

The Points thro’ which the half Hours must pass, may be found by means of the Secants of the odd Hours. For example, the Secant B3, laid off at the Point 3 on the Equinoctial Line, will fall on one side upon the Point for half an Hour past 4, and on the other side, for half an Hour past 10; the Secant B9, will give half an Hour past 7, and half an Hour past 1; B11, will give half an Hour past 8, and half an Hour past 2; B1, will give half an Hour past 3, and half an Hour past 9; B7, will give half an Hour past 6, and half an Hour past 12; and lastly, B5 will give half an Hour past 11, and half an Hour past 5.

The Division of the Equinoctial Line serves to make the Horizontal and Vertical Dials exactly, but chiefly the undeclining Regular Dials, viz. the Polar East and West ones: for there need nothing be added to the facility of constructing Equinoctial Dials, because the Angles that the Hour-Lines make at the Center of the Dials are all equal between themselves.

The Angles that the Hour-Lines of a horizontal Dial make with the Meridian in the Center of the Dial, may be found in the following manner by Trigonometry. As Radius is to the Sine of the Elevation of the Pole, So is the Tangent of the Distance of any Hour-Circle from the Meridian, to the Tangent of the Angle that the Hour-Line of that Hour makes with the Meridian or Hour-Line of 12, on the Horizontal Dial. For example; Suppose the Angle that the Hour-Lines of 1 and 11, make with the Meridian on a horizontal Dial for the Latitude of 49 Deg. be required: form a Rule of Proportion whose first Term let be the Radius 100000; the second, the Sine of 49 Deg. which is 75471; and the third, the Tangent of 15 Deg. (viz. the Tangent of the Distance of the Hour-Circles of 11 and 1 from the Meridian) which is 26597. Now having found the fourth Term 20222, seek it in the Tables of Tangents, and you will find 11 Deg. 26 Min. stand against it: therefore the Angle that the Hour-Lines of 1 or 11 make with the Meridian, is 11 Deg. 26 Min.

Thus may be found the Angles that all the Hour-Lines, and half Hour-Lines, &c. make with the Meridian in the Center of a horizontal Dial, viz. by as many Rules of Proportion, as there are Hour-Lines and half Hour-Lines, &c. to be drawn, whose two first Terms are standing, to wit, the Radius, and the Sine of the Elevation of the Pole; and so you have but the third Term to seek in the Tables; that is, the Tangent of the Hour-Circle’s distance from the Meridian, in order to find the 4th Term. You may take the Logarithms of those Terms if you have a mind to it, which will save the trouble of multiplying and dividing.

The aforesaid Analogy may serve likewise for vertical Dials, if the Sine Complement of the Elevation of the Pole, which is 41 Deg. about Paris, be made use of for the second Term; because any vertical Dial at Paris may be considered as an Horizontal one for the Latitude of 41 Deg.

Moreover, the aforesaid Analogy holds for undeclining Inclining-Dials, if the Sine of the Angle made by the Axis and Meridian-Line at the Center of the Dial be used for the second Term of the Analogy. For Example, Because the Dial B on the Dodecahedron of Figure 2, inclines 63 Deg. 26 Min. you must substract the Elevation of the Pole, which is 49 Deg. from 63 Deg. 26 Min. and then if you make an horizontal Dial for the Latitude of 14 Deg. 26 Min. in taking 14 Deg. 26 Min. for the second Term of the Analogy, you may calculate the Angles that all the Hour-Lines make with the Meridian or Hour-Line of 12.

A Table of the Angles that the Hour-Lines make with the Meridian at the Center of an Horizontal Dial.
Latitude Hours I. and XI. Hours II. and X. Hours III. and IX. Hours IV. and VIII. Hours V. and VII. Hours VI. and VI.
41 Deg. 9D. 58M. 20D. 45M. 33D. 16M. 48D. 39M. 67D. 47M. 90D. 00M.
49 Deg. 11D. 26M. 23D. 33M. 37D. 3M. 52D. 35M. 70D. 27M. 90D. 00M.

## To draw the principal Lines upon a vertical Decliner by Trigonometrical Calculation.

This manner of Calculation consists in the five following Rules.

The Declination of a Plane being given, to find the Angle that the substylar Line makes with the Meridian.

Rule I. As Radius is to the Sine of the Plane’s Declination, So is the Tangent Complement of the Latitude, to the Tangent of the Angle made by the substylar Line and Meridian in the Center of a vertical Decliner. And the Angle that the substylar Line makes with the Horizon at the Foot of the right Style, is the Complement of this Angle. Also the Angle that the Equinoctial Line makes with the Horizon at the Point wherein the Hour-Line of 6 cuts it, is equal to the Angle made by the substylar Line and Meridian; and the Angle of the Equinoctial Line and Meridian is it’s Complement.

Rule II. To find the Angle which the Axis of the Dial makes with the substylar Line, which may be called likewise the Height of the Pole above the vertical Plane; say,

As Radius is to the Sine Complement of the Latitude, So is the Sine Complement of the Plane’s Declination to the Sine of the Angle required. Note, The Angle that the Axis makes with the right Style, is the Complement of this Angle; and the Angle that the Radius of the Equinoctial Circle makes with the right Style, is equal to the Angle that the Axis makes with the Substyle. Also the Angle made by the Radius of the Equinoctial Circle and the Substyle, is the Complement thereof.

Rule III. To find the Arc of the Equinoctial or Angle between the substylar Line and the Meridian in declining Dials; that is, the Difference between the Meridian of the Place, and the Meridian of the Plane, for the substylar Line is the Meridian of the Plane; say,

As Radius is to the Sine of the Latitude, So is the Tangent Complement of the Plane’s Declination to the Tangent of an Arc, whose Complement will be that required.

Rule IV. To find the Angle that the Hour-Line of 6 makes with the horizontal Line, and the Meridian in the Center of the Dial; say,

As Radius is to the Sine of the Plane’s Declination, So is the Tangent of the Latitude, to the Tangent of the Angle that the Hour-Line of 6 makes with the Horizon; the Complement of which, is that made by the Hour-Line of 6 and the Meridian.

Rule V. To find the Angles that the Hour-Lines make with the substylar Line; and by this means, the Angles that they make with the Meridian in the Center of a vertical Dial.

This Proposition is founded upon this Gnomonick Principle, viz. that any Plane may be parallel to some Horizon, and consequently will be an horizontal Dial for that Latitude, the substylar Line being the Meridian, from which the proper Hour-Lines must be laid off on both Sides.

But before this can be done, the Angle that the Substyle makes with the Meridian must be found, by Rule I. the Elevation of the Pole above the Plane, by Rule II. the Arc of the Equinoctial between the Substyle and the Meridian, by Rule III. with the Difference or Degrees of the two first Distances from the Style; one being between the Substyle and the Meridian, and the other between the Substyle and the Hour-Line of 6. These being found, say,

As Radius is to the Sine of the Elevation of the Pole above the Plane, So is the Tangent of the Distance of any Hour-Circle from the Meridian of the Plane or Substylar Line to the Tangent of the Angle made by the Hour-Line of the proposed Hour-Circle and the substylar Line in the Center of the Dial.

Note, If the substylar Line happens to fall upon any half or whole Hour, then the two first Distances of the Hour-Circles from the substylar Line will be each 7 Deg. 30 Min. or 15 Deg. and in this Case, the Angles of the Hour-Lines of the Hour-Circles, equally distant on both sides the Hour the substylar Line falls upon, will be equal on both sides the substylar Line.

## The Application of the precedent Rules to a vertical Decliner of 45 Deg. South-westwardly, the Latitude of 49 Deg. (Vide Figure 9).

The Angle made by the substylar Line and the Meridian, will be found by the first Rule 31 Deg. 25 Min.

The Angle of the Axis and substylar Line, by Rule II will be 27 Deg. 38 Min. and the Arc of the Equinoctial between the Meridian of the Place and the Meridian of the Plane, by Rule III. will be found 52 Deg. 58 Min. and consequently the substylar Line falls between the Hour-Lines of 3 and 4 in the Afternoon; and the Angle made by the Hour-Line of 6 and the Meridian, is 50 Deg. 52 Min.

The Arc of the Equinoctial 52 Deg. 58 Min. being found, substract 45 Deg. which is the Arc of the Equinoctial answering to the Hour of 3, from it, and the Remainder 7 Deg. 58 Min. will be the Arc of the Distance of the Hour of 3 from the Substyle, and consequently 7 Deg. 2 Min. is the Distance of the Hour of 4 from the Substyle.

Therefore to find the Angles that the Hour-Lines make with the Substyle in the Center of the Dial, you must begin with one of these Distances, in saying, for Example, As Radius 100000 is to the Sine of the Elevation of the Pole above the declining Plane, which in this Example is 27 Deg. 38 Min. whose Sine is 46381, So is the Tangent of 7 Deg. 2 Min. which is 12337, to a fourth Number, which shall be found 5722, viz. the Tangent of 3 Deg. 16 Min. and consequently the Angle that the Hour-Line or 4 makes with the Substyle, is 3 Deg. 16 Min. and to find the Angle that the Hour-Line of 5 makes with the substylar Line, you must first add 15 Deg. to 7 Deg. 2 Min. and seek the Tangent of the Sum 22 Deg. 2 Min. and then proceed, as before, and you will find the Angle made by the Hour-Line of 5 with the substylar Line will be 10 Deg. 38 Min. the Angle of the Hour-Line of 6 with the same, will be 19 Deg. 17 Min. the Angle of the Hour-Line of 7, 30 Deg. 44 Min. and the Angle of the Hour-Line of 8 in the Evening, 47 Deg. 25 Min.

But if the Angles that the said Hour-Lines make with the Meridian or Hour-Line of 12 be required, you must add 31 Deg. 35 Min. to each of the aforesaid Angles; and consequently the Angle that the Hour-Line of 4 makes with the Meridian, will be 34 Deg. 51 Min. the Hour-Line of 5, 42 Deg. 13 Min. the Hour-Line of 6, 50 Deg. 52 Min. the Hour-Line of 7, 61 Deg. 19 Min. and the Hour-Line of 8, 79 Deg. 10 Min.

Having calculated, in the abovesaid manner, the Angles made by the Hour-Lines on the other side the substylar Line, with the said substylar Line, you will find the Angle of the Hour-Line of 3, 3 Deg. 45 Min. that of the Hour-Line of 2, 11 Deg. 7 Min. that of the Hour-Line of 1 , 19 Deg. 54 Min. that of the Hour-Line of 12, 31 Deg. 25 Min. that of the Hour-Line of 11, 48 Deg. 54 Min. that of the Hour-Line of 10, 75 Deg. 7 Min. and that of the Hour-Line of 9, 106 Deg. 48 Min.

Now if 31 Deg. 35 Min. viz. the Substyle’s Distance from the Meridian, be taken from each of these last Angles, then the Angle that the Hour-Line of 9 makes with the Meridian, will be 75 Deg. 13 Min. that of the Hour-Line of 10, 43 Deg. 21 Min. that of the Hour-Line of 11, 17 Deg. 19 Min. and so of others.

When the Declination of a Plane is very great, the Center of a Dial cannot then be pricked down conveniently thereon, since the Hour-Lines will fall too near each other. And in this Case they may be drawn between two horizontal Lines; for the Angles that the Hour-Lines make with the said horizontal Lines, are the Complements of the Angles that the respective Hour-Lines make with the Meridian.

## How to find the Declination of an upright or vertical Wall or Plane, by means of the Shadow of the Extremity of an Iron Rod or Style.

Because the Exactness of Vertical Dials chiefly depend on the Knowledge of the Situations of the Walls on which they are to be made or set up against, with respect to the Heavens, that is, their Declinations: therefore it is very necessary that their Declinations be found with all possible Exactness, which we shall endeavour to do before we close this Chapter.

### Preparations.

You must first fix an Iron Rod or Wire in the Wall obliquely, having it’s Extremity sharp and pretty distant from the Wall, as the Rod AI, whose Extremity I is sharp. Vide Fig. 9.

Secondly, The Foot H of the Style must be pricked down upon the Dial Plane. This Point is that wherein the Perpendicular HI drawn from the Extremity of the Rod or Style meets the Plane of the Dial. You must likewise draw the vertical Line HF passing thro’ that Point, which represents the perpendicular Vertical to the Plane of the Dial, and also the horizontal Line DC cutting the said vertical Line at right Angles, in the Foot of the Style H. This being done, measure exactly the Length of the right Style HI or HF, its equal, that is, measure the Distance from the Foot of the Style to its Extremity, with some Scale divided into small Parts. Then having observed where the Extremity Of the Shadow of the Iron Rod falls upon the Wall at different Times in the same Day, as at the Points 2, 3, 4; you must measure the Distance of each Extremity of the Shadow from the horizontal Line with the Scale: as, for example, the Distance from the Point 2 to the Point Z in the horizontal Line; as likewise the Distance from the same Point to the Vertical Line passing thro’ the Foot of the Style; as from the Point 2 to the Point X; and then you must let down the Numbers found orderly in a Memorial, that so they may be made use of in the following Analogies.

But to prick down upon the Wall nicely the Shadow of the Extremity of the Rod or Style, you must use the following Method, which I had from M. de la Hire. Fasten a little Tin-Plate, having a round hole therein, near the Extremity of the Rod, in such manner, that the Extremity of the Iron Rod be exactly in the Center of the said round hole, and the Plate exposed directly to the Sun; then you will see a little Oval of Light upon the Wall in the Shadow of the Plate: and if you draw quickly with a Pencil, a light Track upon the Wall about the said Oval of Light, which is moving continually; the Center of the said Oval may be taken lor the true Shadow of the Extremity of the Rod.

Having thus marked the Points 2, 3, 4, whereat the Extremity of the Shadow falls, you must find the Amplitude, and the Sun’s Altitude answering to each of them, and set them down in the Memorial.

Note, The Amplitude that we mean here, is the Angle that the height of the Style or Rod makes with the Line drawn from each of the observed Extremities of the Shadow to the horizontal Line (for each of these Lines represents upon the Wall the vertical Circle the Sun is in at the Time of Observation). This Angle is marked HFZ in the Figure, and is the Amplitude correspondent to the Point 2. Now to find this Angle, you must say, As the Height of the Rod or Style is to the Distance from the Extremity of the Shadow to the vertical Line, So is Radius to the Tangent of the Amplitude. And by making this Analogy for each Extremity of the Shadow of the Rod observed at different Times, the correspondent Amplitudes will be had, and must be set down in one Column in the Memorial.

Then to find the Sun’s Altitude above the Horizon, you must take the Complement of the Amplitude, and the Distance of each observed Extremity of the Shadow from the horizontal Line. This being done, say, As the Height of the Style is to the Sine Complement of the Amplitude, So is the Distance of the Extremity of the Shadow from the horizontal Line, to the Tangent of the Sun’s Altitude above the Horizon, which being found for the Times of each Observation of the Shadow of the Iron Rod, set them down orderly in one Column.

Note, If the Extremity of the Shadow observed falls upon the vertical Line passing thro’ the Foot of the Style, there will then be no Amplitude; and in this Case you will have the Sun’s Altitude by one. Rule only, in saying, As the Height of the Style is to the Distance of the Extremity of the Shadow from the Foot of the Style, So is Radius to the Tangent of the Sun’s Altitude.

After this, you must find the Distance of each observed vertical or azimuth Line from the Meridian; and in order to do this, the Sun’s Declination must be had for the Times wherein the Extremities of the Shadow were taken: if it be at the time of the Solstices, the same Declination will serve for all the Extremities of the Shadow observed in one Day; but it the Sun be in the Equinoctial, you must have his Declination for each time of the Observation of the Extremity of the Shadow, in taking the Parts proportional.

Now the Sun’s Declination being had, you must take the Complement thereof, as likewise the Complement of his Altitude, and the Complement of the Latitude, and add them all three together; and take half the Sum, and from this half Sum take the Complement of the Sun’s Altitude, and the Remainder will be a first Difference: and moreover, if the Complement of the Latitude be taken from the said half Sum, you will have a second Difference. This being done, say, As the Sine Complement of the Latitude is to the Sine of the first Difference, so is the Sine of the second Difference to a fourth Sine: and as the Sine Complement of the Sun’s Altitude is to Radius, so is that fourth Sine found to another Sine; which being multiplied by Radius, and the Square Root of the Product, will be half the Distance of the Extremity of the Shadow observed, or of its vertical Line from the Meridian or Hour-Line of 12.

This Distance being found in Degrees and Minutes, we may have the Declination of any Wall, which here is the Angle HFE, by some one of the five following Cases.

First, If the Extremity of the Shadow of the Style is between the vertical Line passing thro’ the Foot of the Style, and the Hour-Line of 12, as is the Point 2 in this Example, which was observed some time in the Afternoon; then you must add the Amplitude to the Distance of the vertical Line from the Meridian.

Secondly, If the Extremity of the Shadow falls beyond the vertical Line passing thro’ the Foot of the Style, as here the Point 3 does, you must substract the Amplitude from the Distance of the vertical Line from the Meridian, to have the Declination of the Wall.

Thirdly, If the observed Extremity of the Shadow be found exactly upon the vertical Line passing thro’ the Foot of the Style, then there will be no Amplitude, and its Distance from the Meridian will be the Wall’s Declination.

Fourthly, If the Extremity of the Shadow is on this side of the Meridian, as here the Point 4 is, which was observed before Noon, the Amplitude will be greater than the Declination; to have which, you must substract from the Amplitude the Distance of the Vertical Line from the Meridian.

Fifthly, If the Extremity of the Shadow was observed precisely at Noon, the Wall’s Declination would be equal to the Amplitude; and since the Sun’s Declination, and the Latitude is known, it will be easy to know whether the Altitude observed any Day be the greatest for that Day, that is, whether it be the Sun’s Meridian Altitude. Note, What we have said is easily applicable to all Declinations, whether East wards or West wards, if the Line of Midnight be used instead of that of Noon, when Walls decline North-East or North-West.

An Example will make all this manifest: in order to which, let us suppose, that, in a Place where the North-Pole is elevated, or, which is all one, where the Latitude of the Place is 48 Deg. 50 Min. we have observed the Extremity of the Shadow of an Iron-Rod Upon a very upright Wall about the time of the Summer Solstice, whose Distance from the vertical Line passing thro’ the Foot of the Style is 100 equal Parts of some Scale, and the Height of the Style 300 of the same Parts.

### The Operation by Logarithms.

$$\begin{array}{r} &\text{The Logarithm of 100}\ \ \ \ \ \ 20000000\\ &\text{The Logarithm of Radius}\ \ \ \ 100000000\\ \hline &\text{The Sum}\ \ \ \ 120000000\\ &\text{The Logarithm of 300}\ \ \ \ \ \ 24771212\\ &\text{The Remainder}\ \ \ \ \ \ 95228788 \end{array}$$

This Number remaining is the Logarithm Tangent of 18 Deg. 26 Min. for the Amplitude of the observed Extremity of the Shadow, and the Complement thereof, is 71 Deg. 34 Min.

Then to find the Sun’s Altitude, suppose the Distance from the Extremity of the Shadow observed to the horizontal Line be 600 of the aforesaid equal Parts.

$$\begin{array}{r} &\text{The Logarithm Sine of 71 Deg. 34 Min.}\ \ \ \ \ \ 99771253\\ &\text{The Logarithm of 600}\ \ \ \ \ \ \ 27781512\\ \hline &\text{The Sum}\ \ \ \ \ 127552765\\ &\text{The Logarithm of 300}\ \ \ \ \ \ \ 24771212\\ &\text{The Remainder}\ \ \ \ \ 102781553 \end{array}$$

This remaining Number is the Logarithm Tangent of 62 Deg. 13 Min. the Sun’s Altitude.

$$\begin{array}{r} &\text{Then suppose the Complement of the Latitude is}\ \ \ \text{41 D. 10 M.}\\ &\text{The complement of the Declination of the Sun}\ \ \ \text{66 D. 45 M.}\\ &\text{The complement of the Height of the Sun}\ \ \ \text{27 D. 45 M.}\\ \hline &\text{The Sum}\ \ \ \text{135 D. 42 M.}\\ &\text{Half the Sum} \ \ \text{67 D. 51 M.}\\ &\text{The Complement of the Latitude}\ \ \ \text{51 D. 10 M.}\\ \hline &\text{The first Difference}\ \ \ \text{26 D. 41 M.}\\ &\text{Again, taking from}\ \ \ \text{67 D. 51 M.}\\ &\text{The Complement of the Sun’s Altitude}\ \ \ \text{27 D. 47 M.}\\ \hline &\text{We shall have the second difference}\ \ \ \text{27 D. 47 M.}\\ \end{array}$$

### The first Analogy.

$$\begin{array}{r} &\text{The Logarithm Sine of the first Difference 26 Deg. 41 Min.}\ \ \ \ \ 96523035\\ &\text{The Logarithm Sine of the second Difference 40 Deg. 4 Min.}\ \ \ \ \ 98086690\\ \hline &\text{The Sum}\ \ \ 194609725\\ &\text{The Logarithm Sine of 41 Deg. 10 Min. substract}\ \ \ \ \ 91883919\\ \hline &\text{The fourth Sine remaining}\ \ \ \ \ 96425806\\ \end{array}$$

### The second Analogy.

$$\begin{array}{r} &\text{The Logarithm of Radius}\ \ \ \ \ \ 10000000\\ &\text{The fourth Sine}\ \ \ \ \ \ 96425806\\ \hline &\text{The Sum}\ \ \ \ 196425806\\ &\text{Substract the Logarithm Sine of 27 Deg. 47 Min.}\ \ \ \ \ \ 96685064\\ \hline &\text{The remaining Sine}\ \ \ \ \ \ 99740742\\ &\text{The Log. Sine of Radius}\ \ \ \ 100000000\\ \hline &\text{The Sum}\ \ \ \ 199740742\\ \hline &\text{The half of this Number for the Square Root}\ \ \ \ 998870371\\ \end{array}$$

This last Number is the Logarithm Sine of 76 Deg. 4 Min. which being doubled, makes 152 Deg. 8 Min. but since this Angle is obtuse, you must substract it from 180 Deg. and the Remainder 27 Deg. 52 Min. is the Distance of the observed vertical Circle or Line from the Meridian: and because the Extremity of the Shadow 2, for which the Calculation is supposed to be made, is between the vertical Line passing thro’ the Foot of the Style, and the Hour-Line of 12; you must add the aforesaid 27 Deg. 52 Min. to the calculated Amplitude 18 Deg. 26 Min. to have the Declination 46 Deg. 18 Min.

The Declination of a Wall may be found by one Observation of the Extremity of the Shadow of a Style or Iron-Rod only; but it is better to make several Observations thereof in one Day, or in different Days, that so the Declination of the Wall may be calculated for each Observation, and the proportional Parts of the Differences arising may be taken: if, for Example, the Extremity of the Shadow of the Style hath been six times observed, you must take the one-sixth Part of the Differences produced by the Calculations, in order to have the true Declination of the Wall.