PROPOSITION IX.

The Day being given, to find where the People in the Frigid Zone will be Perifcii; and that the first Day.

FIND the Places where the Sun begins not to fet; and you have the People there.

PROPOSITION X.

Thofe under the Equator have the Shadow one half of the Year to the North of them, and on the other half to the South of them and on the two Days of the Equinox they are Afcii, or without Shadow.

THIS needs no Explanation.

PROPOSITION XI.

To place an horizontal Plane fo as an upright Style on it have no Shadow fome Days; and on other Days project one either to the North or South, as it is with thofe in any given Place of the Torrid Zone.

SUBSTRACT the Latitude of that Place, from the Latitude of your Place (but add if of different kinds), and what Degrees remain, make your upright Style bend fo many fouthward from the Perpendicular; and the Plane to which the Style is perpendicular, bending along with it, will be the Plane on which Shadows will be projected as on the horizontal Plane in that Place of the Frigid Zone.

PROPO

PROPOSITION XII.

In Places under the Equator the Shadow of an upright Style is in one right Line, all the equinoctial Days, being on the West Side of the Style in the Forenoon, and on the East Side in the Afternoon; but on other Days the Shadow turns in a Semicircle.

IN Places in the Torrid Zone, while the Sun is between the Tropic and the Parallel in which the Place lies, the Shadow describes lefs than a Semicircle; and in Places in the Temperate Zone, while the Sun is in the South Part of the Ecliptic, the Shadow defcribes alfo a lefs Space than a Semicircle; but in the North Part a greater Space, and in the Equinoxes the Shadow is carried thro' a Semicircle, excepting in the Equator, and at the Pole.

THESE may all appear by inspection on the Globe; or by defcribing a Figure.

PROPOSITION XIII.

In Places in the Torrid Zone, while the Sun is between the Parallel of the Place and the Tropic nearest to it, the Shadow then goes back twice, repeating the Directions to the fame Points, once in the Forenoon, and once in the Afternoon: and the Sun will then feem to go back.

ELEVATE the Globe to a Latitude less than thirty three Degrees thirty Minutes; and defcribe the Parallel of that Latitude. I fay while the Sun is between that and the Tropic, the Sun and the Shadow will seem to go twice back; and to repeat the fame Directions. Apply the Quadrant of Altitude to the Zenith, and turn it about 'till it touch

the

the Parallel, in which the Sun is drawn on the Globe; and bring the Point of Contact to the Eaft of the Horizon.

From the Sun in that Point draw a Line to the Zenith of the Place, and it will cut the Line of the Sun's Courfe in a Point higher above the Horizon, where the Sun will direct the Shadow the fame way as when it was rifing; having had other Directions repeated between thefe two. And the fame will be on the Weft Side of the Sun's Courfe, where it will fet in the fame Point it was in a few Hours before.

THIS will appear alfo in Fig. 32. Suppofe a Place in the Torrid Zone, as L, in which there is a Style erected; and let AMF be the Tropic, or the Sun's Parallel; and let the Sun be rifing in A: then the Shadow of the Style is projected in the Line La; and when the Sun comes to C the Shadow will be in the Line Lc, and when it comes to G it will be again in the Line La, and when it comes to the Meridian in M the Shadow is directed South in the Line L m, and when it is in E, and fets in F, it will again have the fame Direction.

COROLLARY

'TIS no Miracle that Shadows go back on Dials, except they go back on a fudden; or if they Point back to the fame Hour-Lines, if the Style be not perpendicular, but parallel to the Poles: and tho' it be perpendicular, yet the Line of Shadows doth not fhow the Hour, except the Plane of the Dial be in the equatorial Plane.

VOL. II.

H

PRO

PROPOSITION XIV.

A Plane being given in the Torrid Zone, and one of thofe Days in which the Sun and Shadow feem to go back to find the Point in which the Sun then is, and the Hour when it will be.

ELEVATE the Globe for the Latitude, and mark the Sun's Place in the Ecliptic, and draw with Chalk it's Parallel that Day; and applying the Quadrant to the Zenith, turn it about 'till it touch the Parallel, and it will be on that Point. AND to find the Hour, obferve the Point of Contact, bring the Index to 12 on the horary Circle, and bring the Point to the Meridian, and the Index will show how many Hours the going back begins before or after Noon.

PROPOSITION XV.

The Lengths of the Shadows decrease as the Sun rifes above the Horizon; and vice versâ, increase as the Sun goes from East to the South; and again decrease as it goes from South to West.

FOR the Sun, the higher it is, comes nearer to the Zenith of the horizontal Plane; and therefore the Ray that terminates the Shadow comes nearer the Style of the Dial; and the Shadow thus becomes fhorter: and the Sun being at the greatest Height on the Meridian, the Shadow must be fhorteft; and in the Horizon it hath no Altitude; and therefore the Shadow is infinite.

2.

PRO

PROPOSITION XVI.

Having the Altitude of the Style and Length of the Shadow, to find the Sun's Altitude; and from thence the Hour; if the Latitude and Day of the Month be known.

THE Length of the Style, the Shadow, and the Ray that terminates the Shadow, make a right angled Triangle; and according to Prop. xv. Chap. ii. make this Proportion: As the Length of the Shadow is to the Length of the Style, So is the Radius to the Tangent of the Angle of the Sun's Altitude. Then find the Hour by Prop. iii. Chap. xxix.

PROPOSITION XVII.

Having the Semidiameter of the Sun and Earth, to find the Length of the Shadow which the whole Earth cafts in the Heavens.

THE Shadow of the Earth is conical, as they that are skilled in Optics demonftrate; and may be easily fhewn by a Figure. Therefore we are to find the top of the Cone, (which comes on the Moon when eclipfed) how far it is from the Center of the Earth; thus: As the Difference of the Semidiameter of the Sun and Earth, is to the Sun's Distance from us; So is the Semidiameter of the Earth, to the Length of the Earth's Shadow, or of the Axis of the conical Shadow,