The English globe being a stabil and immobil one, performing what the ordinary globes do, and much more / invented and described by the Right Honorable, the Earl of Castlemaine ; and now publish't by Joseph Moxon ...

OPERATION XXXIII. To make a compound Dial to wit, one containing several useful Operations.

INnumerable are the ingenious Dials that may be invented, but since we have been long enough on this Subject, either for my Reader's Speculation or Curiosity, I will now conclude, and that with a Recapitulation or summing up of much of what we have already said, by showing the Fabrick of a Com∣pound Dial; that is to say, one that contains many useful Ope∣rations, besides the Hour; for nothing rubs up the Memory more efficatiously, or makes us more Masters of our Rules, than a Practical Example.

The said Dial shows as follows.

• 1. The Hour with us at all times.
• 2. The Hour in what other Countries you please.
• 3. The Sun's Place in each Sign.
• 4. The Day of the Month.
• 5. The time of the Sun's Rising and Setting.
• 6. The Sun's Amplitude.
• 7. The Sun's Height.
• 8. The Sun's Azimuth.
• 9. The Sun's Bearing according to the Points of the Com∣pass.
• 10. The Proportion between Perpendiculars and their Sha∣dows, and consequently the height of any Tower or the like.

To make then this Dial, you must first describe an Hori∣zontal* 1.1 (as in Sch. 41.) about a Foot in Diameter, and let B the Center of the Plane be the Point, where an Erect, or Ʋpright Stile (according to our Directions in the * 1.2 first Horizontal;) shews you with its Top the Hour. Now because the Shade of an Ʋpright Stile, unless it be very short, will presently fall out of the Plane, as well in the Morning as toward Night,

therefore it will be convenient to have your Cock or Stile made so, that AB the Perpendicular or fore-part of it (as in Scheme 42.) should stand at B the said Center of the Plane, to repre∣sent this upright Stile, and its Angle AOB at O the Cen∣ter of the Dial, or Point from whence all the Hour-lines are drawn; for thus the side OA (making with the Meridian line at O, the Angle of the Elevation) represents the Axis of the World, and consequently casts its shadow on the Hour-lines, as the usual Cocks of all Horizontal Dials do.* 1.3

2. Having chosen all the Places, which you desire from time to time to know what a Clock it is at, consider well your Globe, and find under what Hour-Circles the said Places lye; as for Example, suppose Rome lies under the 11 a Clock Hour-Cir∣cle, Constantinople under that of 10, Aleppo 9, &c. Place therefore the said Towns towards the Limb of your Dial, un∣der the corresponding Hour-lines, and you will constantly know the time of the Day in the said Places; for calling it always Noon at each Place you seek after, you have nothing to do but to count the Hours from thence to the shade of the Stile; as v. g. If it be 4 a Clock with you in the afternoon, and you would know the Hour at Aleppo, let Aleppo be 12, and counting from thence (1. 2. 3. &c.) 'till you come to the Hour of the Day, (I mean the Hour then shown you by the Shade,) you will find it to be 7 a Clock there; for Aleppo is (you see) three hours Eastward of you; now had the Hour with you been 4 in the mor∣ning, you must have counted backwards, as 11, 10, 9, 8, and consequently you would have found it there 8 in the morning. In this manner then you must operate all along.

3ly. and 4ly, Find by your Globe exactly the Sun's height e∣very* 1.4 hour at his Entrance into each Sign, then take by the help of your Sector (AB, the Erect Stile in Scheme 42. being Radi∣us) the Tangent Complements of the Heights, and putting one Foot of your Compasses on your Dial at B, make Pricks or Marks in each corresponding Hour-line accordingly; that is to say, if the Sun be high (suppose) 50 Degrees at 12 of the Clock, when he enters ♉ or ♍, then take the Tangent of 40 and prick that distance in the Meridian line, viz. From B to f; and if his height at 1 and 11 a Clock be (v. g.) 48 degrees, take the Tangent of 42, and prick that distance in the 11 and 1 a Clock lines, viz. from B to h and g, and when you have gone thus o∣ver

all the Hour-lines, no sooner will the Sun come into ♉ or ♍ but the Shade of the Point or Apex of the Stile AB will fall e∣very hour on the aforesaid Pricks, and consequently show you the Suns place in the Ecliptic. In like manner you must do with the rest of the Signs, and then with the 10th Degree of eve∣ry Sign, placing still the Character of each Sign about the Limb of your Dial, near the last mark or Prick belonging to it. This being done, see by your Globe what day of the month corresponds with each Sign, and what with their Subdivisions, and if you mark this (as the said 41th Scheme shows you) on both sides of the Meridian, then the said Pricks will (by the help of the Shade of the top of AB) show you also the day of the month. I mention here Pricks not only as an easier way, but a better way than Lines; for besides the great difficulty of draw∣ing them, they embarras and confound a Dial very much, es∣pecially if there be many of them; whereas the said Pricks are never out of an Hour-line, and consequently take up no new room. Now to avoid Confusion and Mistakes, I would have the said Pricks of 3 sorts at least, for if one Row were (v. g.) Astericks and another Crosses, and a 3d Plain Pricks, you would then know at first sight, to what Sign or Day of the month any of them belongs.

5ly. Instead of troubling you with deviding the Circle* 1.5 GKLT (the upper part of the Border of the Dial) for the finding out the time of the Suns Rising and Setting, you need only consult the Days of the Month on your Globe, first, when He rises earliest, Secondly, when He rises at 4 a Clock, Third∣ly, when at 4½; Fifthly, when at 5; and in the like Proportion go on, till the Days come to their greatest Decrease, and put∣ting the said days of the Month in Order (as they are in the Scheme) under the corresponding Hours on the morning side of your Dial for his Rising, do the like for his Setting on the Evening side of it, and you may perform the Operation with sufficient Exactness. In like manner you are to proceed for the Quarters, half Quarters, &c. if you would have them exprest.

6ly. To avoid also the trouble of deviding the Circle* 1.6 〈 in non-Latin alphabet 〉〈 in non-Latin alphabet 〉 according to the Suns Diurnal Increment and Decre∣ment in Amplitude, you need only find by your Globe, what the said Amplitude amounts to on every of the aforementioned Days (which are markt on your Dial for the Suns Rising and

Setting) and then put it in Figures under each Day, as the Scheme shows you.

7ly. Open your Compasses at the Tangent of 28 Degrees (AB* 1.7 being the Radius) and putting one Foot on B describe the Cir∣cle XYZ, afterwards describe another according to the Tan∣gent of 35 Degrees, then a third, according to that of 40, and so on in the same Proportion as far as your Plane permits. Now if you mark these Circles with the Figures of the Comple∣ment of their Degrees, that is to say the Circle of 28 Degrees with the Figure 62, that of 35 with 55, that of 40 with 50, &c. you will always know the height of the Sun; for what Circle so∣ever the Shade of AB touches with its Top, that will be the requir'd Height; and if it falls between 2 Circles, 'tis but con∣sidering which of them it comes nearest to, and then you may guess at the Height with sufficient exactness.

8ly, and 9ly. Devide one of these Circles viz. SEWN into* 1.8 Degrees, and under each 11 Degree and ¼, place the several Points of the Pixidis Nauticae, or Mariners Compass in the Or∣der as they are express'd in our said Scheme, and you will not only have (by the Shade of AB) the Suns Azimuth at all times, but see also how he bears from you according to the Points of the Compass; and if the Shade be at any time too short, lay on it but a Ruler, Label of Paper or the like, and that will truly lengthen the said Shade, and resolve your Question.

10thly. Devide AF the Northern half of the Meridian, as* 1.9 many times as you can by the Stile or Radius AB, and then each Devision into ten equal parts (as you see it done in the said Scheme) and by it you will know at all times the Propor∣tion between any Perpendicular and its Shade, and consequent∣ly, (besides many other things) the height of any Tower, Tree or the like, for having found the Sun to be (suppose) 25 De∣grees high, and that the Circle of Altitude cuts the Linc AF in the 22 Devision, if therefore you measure the Shade of your Tower, and finding it (for Examples sake) to be 66 Yards long, you have what you seek; for as the said 22 is to 10 (the Stiles height) so is 66 the length of the Shade to 30 the height of the Tower.

So much then for the Construction of Dials. And now let me desire all those that are pleased to follow this Geometrical way (which perchance is as expedite a one, and as free from blind

Lines as can be,) not to rest satisfy'd till they fully comprehend what they do; for the Mechanical way of Dialling is as soon lost as learnt, it being impossible (without continual Practice) not to forget the Rules, especially if one can make many Dials; when as a man that understands the reason of the Operations (by having in his Head a true Idea of the Sphere and its Proje∣ction) will 20 years after without Memorandums or Notes, be able (reflecting but a little) to make not only all Dials he for∣merly knew, but new ones also at first fight.

To Conclude, I here present my Reader with the Globe in a new Dress, for being painted or stain'd on Marble (according to Sch. 43.) 'twill be fit for any Garden or open Portico; and least it might appear too plain, the corners of its Base or Pedestal may be adorned with handsom well turn'd Branches, which not only embellish the whole Machin by their Make, But hold out Bowls of Glass and Wyar for use also.

For on the First Corner, to wit, That markt with A, there is* 1.10 placed (as a Rarity.) The blind man's * 1.11 Dial. On the Second markt with B. The † 1.12 Dial that shows the Hour, when the Sun shines not, which will be often very useful. On the third, mark't with C, there is an Armillary Wyer Sphere having a Vane on the Top, that continually shows on the brass Plane within (graduated and Nautically Character'd) from what Quarter the Wind exactly blows; as also, (if you turn the said Vane into the Plane of the Sun) his Azimuth and Bearing. Besides, the Sphere (being an Horizontal Concave Dial) shows the Hour too; for the Shade of the Pin's top in the Center ever fall's on the true Hour-Circle, as I show'd in the * 1.13 Construction of such a Dial. And by the way you must know this Branch stands not in it's true place in the Scheme; I mean on the third Corner of the Base, because in Perspective 'twill fall on the Globe it self, and conse∣quently not appear well to the Eye in a Picture. Lastly, on the fourth Corner markt with D there is another Glass Bowl of the former Dimension, containing orderly all the Constellations, and remarkable Stars, and therefore, if you know the Hour, it will compose the said Bowl or Globe, and so represent the then position of the Heavens; but (tho you are Ignorant of the Hour) if you see a known Star, and move the Bowl on its Axis, till the painted star on it lyes just between your Eye and the Real one, you have the Hour, and consequently may know (the Globe being

now Compos'd) any Star or Constellation above the Horizon; for the Axis of this Bowl having one end pointing directly to the North Pole, and the other fixt in the Center of a Rundle containing on its Limb the Days of each Month, fitted to the right Ascension of the Stars, and moving also on a Plane divided into 24 equal parts, figured with the hours of a Natural Day, 'twill follow that the Day of the Month (when the Globe is Compos'd) must lye on the true Hour, as the true Hour move'd to the Day of the Month must Compose the Globe, as is before hinted. These short directions are sufficient for any Mathematician, or In∣strument-Maker; and as for the Branch it self, 'tis (as you see) not in its true Place for the above mentioned Reason.

J. Moxon To the Reader.

HAving Courteous Reader * 1.14 engaged to show you the Pro∣blems and Operations on the Sector, which the Noble Au∣thor supposes every one (that studies the Geometrical way of Dialling) to know, I shall here begin.

I. Upon a Line given (AB) to erect (CD) a Perpen∣dicular.

[illustration]

IF there be a Point (as C) given in (AB) the Line on which the Perpendicular is to fall, Mark on both sides of the said Point (with your Compass) the equidistant Points M and N, then opening them at pleasure, put one foot on M and describe the blind Arch EF, and putting the other Foot in N, describe the blind Arch GH, and the fair line from (D) their Intersection to the Point C, will be the Perpen∣dicular requir'd. Now if you have no Point assign'd (in the said Line (AB) to terminate your Perpendicular by take two Points there at pleasure, as suppose M and N, and opening how you will your Compasses, describe the blind Arches EF and GH above your Line, and OP and QR below it, and the In∣tersections

of these Arches (to wit, D and S) will be two points to draw your Perpendicular by.

II. Upon (C) the end of (AC) a given Line, to draw (DC) a Perpendicular.

[illustration]

OPEN your Compasses at a convenient width, and putting one Foot on C, let the other (within reach of AC) mark any where, as at F: then touching or cut∣ting from thence the said AC (with the moving Foot of your Compasses) at, suppose, E, and describing on the other side of F the blind Arch GH, lay your Ruler on FE, and it will cut the said Arch, at, suppose D, so that DC will be the requir'd Perpendicular.

III. A Line (AB) being given how to draw (DG) a Pa∣rallel to it.

[illustration]

HAVING taken two points in the said Line, as suppose A and B, open your Compasses at what width you please, and putting one foot on A, describe the blind Arch CDE, and putting one foot on B describe the blind Arch FGH, then if you lay your Ruler on the highest part or greatest Extuberancy of the said Arches, to wit on the Points D and G, the Line so drawn will be the requir'd Parallel.

IV. To describe a true Square.

[illustration]

AB being a Line as long as the side of the Square you design, erect on the end A, the Perpendicular DA of the for∣mer length; then taking between your Compasses the said AB, put one foot on D, and describe the blind arch EF, and again putting one foot on B, describe the blind arch GH, to cut EF, and if from their Intersection C, you draw the fair lines CB and CD, you have a true Square.

V. To draw an Oblong, or (as they commonby call it) a Long Square.

[illustration]

AB being the longest side of this Square, erect on the end A, the Perpedicular DA, of the length of the shortest; then taking between your Compasses, the line AB, put one foot on D, and describe the blind arch EF: and taking between your Com∣passes the line AD, describe the blind arch GH, to cut the said EF, and if from their Intersection C, you draw the fair lines CB and CD, you have the Square you design.

VI. To Describe an equilateral Triangle, or an Isos∣celes.

[illustration]

OPEN your Compasses at AB, being the side of the Triangle you design, and putting one foot on A, describe the blind Arch EF, and again putting one foot on B, describe the blind Arch GH to cut the said EF, and if from their In∣tersection C, you draw the fair lines CA, and CB, you have a true equilateral Triangle; Nor is there any difference in the Description of the Isosceles ASB, for the only difference be∣tween them is, that the sides AS and BS of the Isosceles are longer (or if you please they may be shorter) than the Base AB, whenas all three sides are equal in the equilateral Triangle.

VII. To make a Triangle of three given Lines.

[illustration]

SUPPOSE the first line given be AB, the second AC, the third BC, and that you are to make a Triangle of them: let AB be the Base, and taking the given line AC between your Compasses, put one foot on the Base at A, and describe the Blind Arch EF, then taking the given line BC, between your compasses, put one foot on the Base at B, and describe the Blind Arch GH, to cut the said Arch EF, and if you draw lines, from their Intersection at C, to A and B, on the aforesaid Base, you have your intent.

VIII. To describe an Oval.

[illustration]

CROSS RP at right Angles with IM, and taking with your Compasses (on the said lines from the intersection O) equal distances, to wit, OA, OB, OC, and OD, and draw through the point C, the lines AK and BH, each equal to twice AC, as also throu' D the lines AN and BL, each equal to twice BC, then A and B being Centers, describe the Arches KPM, and HLR; in like manner C and D being Centers, describe the Arches HIK, and LMN, and the figure thus drawn will be a perfect Oval.

[illustration]

* 1.15So much for the Geometrical Problems necessary for Dial∣ling, and as for the Instrumental ones, i. e. those performed by the Sector, they are, as I may say, of two sorts, some belonging to one side of it, and some to the other; for the side marked with L is divided into 100 equal parts, and called the LINE of LINES, and the side mark'd with S, the LINE of SINES. First then of the LINE of LINES, which by the way, tho' it be divided (as I said) but into 100 parts, may yet stand for 1000, if you fancy every 10 Divisions a Line of 100 parts, and in like manner it will stand for 10000 parts, if every division be deemed 100, therefore a Line (v. g.) of 75 equal parts, may be exprest by 75 of those Divisions, or by 7½ or by ¾.

The Use of the LINE of LINES marked with L.

I. To divide a Line into any number of equal parts.

SUPPOSE your Line were to be divided in 23 equal parts, take it between your Compasses, and opening your Sector, place one foot of your said Compasses on the 23 division of the Sector, and the other foot on the 23 over against it, and the distance between the Figures 1 and 1▪ on the said Sector will give you one equal Division of your Line, and the distance between 2 and 2, will give you two equal Divisions of it, and in this man∣ner proceed till you quite run over it, as you design.

II. To find the proportion between any two Lines.

SET over the greater Line at 100, and 100 on the Sector, then taking the lesser between your Compasses, find where it will be just set over also, or lye parallel to the former, which hapning suppose at 50 and 50, you may conclude, that the Proportion re∣quired, is as 100 to 50.

III. To divide a Line as any other Line proposed is di∣vided; that is to say, according to any Proportion.

SUppose you saw a Line, containing 65 equal parts of the Sector devided into three pieces, the first containing five equal parts of the Sector, the other fifteen, so that the last must be 45; then suppose you would divide (after this proportion) another Line, containing but thirteen equal parts of the Sector; Open your Compasses at 13, or length of the Line to be devi∣ded, and putting it over at 65, and 65 on the Sector, the Pa∣rallel at 5 and 5 will be the first division of the Line to be divi∣ded, and one equal part of the Sector in value; the Parallel at 15 and 15 will be the second, and three equal parts in value; and the remainder (being 9 in value,) will be the third; and thus you may do in all other cases.

IV. To encrease or diminish a Line in any Proportion.

SUppose the Proportion were as 4 to 7, take the Line given between your Compasses, and setting it over on your Sector, at the Figures 4 and 4, the distance from 7 to 7 will be a Line encreast (in respect of the given one) as is the Proportion of 4 to 7; In like manner you must do, if any other Proportion were requir'd.

Now if you would diminish a Line as is 7 to 4, put over the Line given at 7 and 7, and the distance of 4 and 4 is the re∣quir'd proportion.

V. Two Lines being given, to find a Third Proportional.

FIND by your Compasses how many parts of the equal Di∣visions of your Sector will measure both your given Lines, so that supposing the one to contain 10 parts, and the other 20, set the second Line (i. e. the Line 20) over at 10, and 10 on the Sector, and the distance or Parallel at 20, and 20 on the Sector, will be 40, the requir'd Proportional.

VI. Three Lines being given, to find a Fourth Proporti∣onal.

THE value of the Lines being found as before, and supposing the first to be 10, the second 20, the third 30, put over the second Line (to wit, 20) at 10 and 10. (the value of the first Line) and the distance or Parallel at 30 and 30. (or value of the third Line) will be 60, the required Proportional.

Of the Use of the LINE of SINES, markt with S.

[illustration]

I. How to find the Sine of any Angle, according to any Radius.

SUppose the Sine of the Angle you require be 50, take the Radius between your Compasses, and put it over at the extremity of the Sector, that is to say, at 90 and 90, and the Parallel at 50 and 50 will be the Sine of 50 Degrees, accord∣ing to that Radius.

II. How to find the Chord of any Arch.

SUppose you would have the Chord of an Arch of 50 Degrees, open your Compasses at the length of the given Radius, and put it over at 90 and 90, then take with your said Com∣passes the Parallel, at the Figures 25 and 25 on the Sector, (i. e.

at the Figures of half the Degrees given) and prick or measure it twice upon any streight Line, and that will be the required Chord.

III. How to make an Angle of any value, as also how to find the value of any Angle already drawn.

[illustration]

SUppose you are to make an Angle of 50 Degrees, draw a Line, as (for Example sake) AD, and taking any Point in it, as B, open your Compasses to a convenient Radius, and put one Foot on B, and describe the blind Arch CF, then taking between your Compasses the Chord of 50, according to the Radius of the said Arch, put one Foot on C, and the other marking at suppose E, draw the Line BE, and you have the required Angle, to wit, the Angle EBC. But if the Angle you would make be above 90 Degrees, as suppose 130, make the Angle of its Supplement, viz. the Angle of 50 as before, and the Angle on the other side, viz. EBA will be the Angle you look for:

Here therefore you see how to find the value of any Angle already drawn, as suppose the Angle EBC, since 'tis but de∣scribing a blind Arch, as CF, and setting over (on the Sector) the Radius of the said Arch at 90 and 90; for if you observe where the measure of this Arch (viz. C. E) marks a Parallel on the Sector, as before, the Figures there (to wit, 25) being doubled (and amounting consequently to 50) will be the requir'd An∣gle.

As for the Tangent and Secant of any number of Degrees, the Noble Author has himself showd you how to find them at pag. 71. and so gentle Reader having finish'd my promise, I bid you Farewel.

Notes

• * 1.1

  The Constru∣ction of it, as to the Hour at home.
• * 1.2

  pag. 73.
• * 1.3

  The Constru∣ction of it as to the Hour in other pla∣ces.
• * 1.4

  To find the Suns place and day of the Month.
• * 1.5

  To find the Rising and Setting of the Sun.
• * 1.6

  To find the Suns Ampli∣tude.
• * 1.7

  To find the Height of the Sun.
• * 1.8

  To find the Suns Azi∣muth and Bearing.
• * 1.9

  To find the Proportion of Perpendicu∣lars to their Shades.
• * 1.10

  The descripti∣on of the Bran∣ches or Embel∣lishments in Sheme 43.
• * 1.11

  p. 111.
• † 1.12

  p. 112.
• * 1.13

  vid. p. 113.
• * 1.14

  p. 71.
• * 1.15

  Of the Sector.