To make an erect dial facing the north I Invert the whole of that just described making the gnomon point upwards i • stead of downwards, and causing all th lower points to be transferred from lef • to right, and from right to left. kind of dial will skew the hours befo VI. A. M. and after VI. P. M. When sue is wanted, the best way is to set up stout post, with the planes of two di back to back, they pointing due and due north, respectively ; thus, as pin retires from one, it will set upon other.
We shall now instruct the read how to make those scales, which indispensable towards the attainment perfection in this pleasing branch • study.
The lines useful in dialling are, 1, ' line of chords ; 2 a line of latitudes ; 3, ' a line of sines; and 4, a line of hoursi They are all derived from the of a circle, as will be shewn in fig. 4.
Describe a circle aid civieleafrito four equal parts by the lines M13,,and C D, in tersecting in the centre E. Draw the chords A C, C B, B D. Now divide the two segments or quadrants, A D and C B, each into nine equal parts ; either of which contains ten degrees. Placing one leg of your compasses at B for a centre, draw the several arcs from the quadrant subtended by the chord C B, so that they may fall upon that chord, which being numbered according as the several arcs correspond with the division on the qua drant, will give a line of chords gradu ally diminishing from B towards C : all the intermediate degrees or the mea= sures of 10° each, thus obtained, may be removed in the same manner from the quadrant, if it be graduated accord ingly.
It will be proper to observe, in this place, that the chord of 60° is the radius of a circle whose quadrant is.subtended by 90° of the same scale : hence, a line of chords is easily made upon any circle, so that any part of that circle may be cut off at pleasure. This is essential every branch of mathematics ; but in dialling it is indispensable to be known : the reader will have observed, that in forming the horizontal dial, the hour lines are drawn thrugh particular points, so as to make the required angles. As he may be at a loss how to effect this on many oc casions, we shall give au example in fig 5, whereby every doubt or difficulty will be removed.
Let it required to cut off an angle of forty degrees from the quadrant, which appertains to a circle for which we have not a line of chords in readiness. On the base line A B measure sixty degrees from any line of chords you may have at hand: it may either exceed or be less than your base line ; we will suppose the former : in this case the base line must be pro longed to the measurement of 60° from your scale, which will carry it on to C. With that 60°, as a radius, and from A, as a centre, describe the quadrant C D, concentric with the quadrant E B, from which you would cut off 40°. Now mea sure 40 degrees on your line of chords, and, placing one foot of the compasses at C, carry the measurement to F, which will cause the angle F A C to measure 40°, and the line F A will, atlC, cut off 40 degrees from the quadrant E B. For an angle does not vary by prolongation ; therefore, if the exterior quadrant is cut at 40°, the interior quadrant, being eon centric therewith, must correspond with that division. • We now proceed to the opposite qua drant, which is not subtended by a chord, but is divided into nine equal parts often degrees each. Draw from the several
points of division on the quadrant eight lines, all parallel with E A, and falling on the radius E D ; this gives a line of sines • which is of very extensive use in various branches of mathematics. From A draw eight lines, passing through the several points ascertained on the line of sines, to the quadrant B D : these will cut the chord subtending that quadrant, and give thereon a line of latitudes, of equal length with the line of chords, hitt very differ ently divided.
The remaining quadrant C A is to be divided into six equal parts, viz. of 15° each : make the chord C F A, and draw its parallel tangent G H. Through the several points of division on the quad rant, draw lines from the centre E to the line G H, which will then represent a line of hours : one of the extremes will be XII, the other will be V1 ; the seve ral intermediate places of I, II, HI, IV, and V, being ascertained by the various lines proceeding from E.
The 6th figure shews part of a dial, constructed by means of the lines of lati tudes and of hours. Having set off the parallels for the substile, and drawn the line of Vi o'clock, set off the• latitude of your place from A towards B; taking the measurement from the line of latitudes. Then measure the whole extent of your line of hours, and. placing one leg of your compasses at B, let the other fail wherever it may reach on the line C A. Divide the line B C according to the mea sures on your line of and from A draw lines through the points of division to the hour circle, which will thus be tru ly intersected at the Loral points. We have before stated, that by dividing the quadrant C A, in fig. 4th, more minutely, that is, by dividing each of the six por tions into four, the halves and quarters of hours may be shewn.
. Having already the modes of con structing those dials which are in ordina ry use, we must refer the more curious reader to Ferguson's " Lectures," for a great variety of dials, which could not be introduced into this work without greatly augmenting the volume. He will there find the modes of constructing dials by logarithms, and by trigonometry; togeth er with many items relating to the more abstruse parts of our subject. We shall briefly add, that the following general principle governs the formation of all dials. Take the words of that great lumi nary of mechanics, the late James Furgu son, F. It. S.
" If the whole earth were transparent and hollow, like a sphere of glass, and had its equator divided into 24 equal parts by so many meridian semicircles, one of them being the geographical meridian of any given place, say London ; and if the hours of XII. were marked on the equa tor, both on that meridian, and on its op posite one, and all the rest of the hours on the rest of the meridians, those meri dians would be the hour-circles of Lon don : then if the sphere had an opaque axis, terminating at its poles, the shadow of that axis would fall upon every particu lar meridian and hour, when the sun came to the plane of the opposite meri dian'; and would, consequently, spew the time at London, and at all th'e other pla ces on the meridian oiLondon."