The semicircle on a sector in two books. Containing the description of a general and portable instrument; whereby most problems (reducible to instrumental practice) in astronomy, trigonometry, arithmetick, geometry, geography, topography, navigation, dyalling, &c. are speedily and exactly resolved. By J. T.

CHAP. III.

Some uses of the Line of natural signs on the Quadrantal side of the fixed piece.

PROBL. 1.

How to adde one sign to another on the Line of Natural Sines.

TO adde one sine to another, is to aug∣ment the line of one sine by the line of the other sine to be added to it. Ex. gr. To adde the sine 15 to the sine 20, I take the distance from the beginning of the line of sines unto 15, and setting one point of the Compasses in 20, upon the same line, turn the other toward 90, which I finde touch in 37. So that in this case (for we regard not the Arithmetical, but proportional aggre∣gate) 15 added to 20, upon the line of na∣tural sines, is the sine 37 upon that line, and from the beginning of the line to 37 is the distance I am to take for the summe of 20 and 15 sines.

PROBL. 2.

How to substract one sine from another upon the line of natural sines.

The substracting of one sine from an∣other, is no more than taking the distance from the lesser to the greater on the line of sines, and that distance applyed to the line from the beginning, gives the residue or re∣mainer. Ex. gr. To substract 20 from 37 I take the distance from 20 to 37 that apply∣ed to the line from the beginning gives 15 for the sine remaining.

PROBL. 3.

To work proportions in sines alone.

Here are four Cases that include all pro∣portions in sines alone.

CASE 1.

When the first term is Radius, or the Sine 90.

Lay the thread to the second term coun∣ted on the degrees upon the movaeble piece from the head toward the end, then num∣bring

the third on the line of sines, take the nearest distance from thence to the thread, and that applyed to the Scale from the beginning gives the fourth term. Ex. gr.

As the Radius 90 is to the sine 20, so is the sine 30 to the sine 10.

CASE 2.

When the Radius is the third term.

Take the sine of the second term in your Compasses, and enter it in the first term up∣on the line of sines, and laying the thread to the nearest distance, on the limb the thread gives the fourth term. Ex. gr.

As the sine 30 is to the sine 20, so is the Radius to the sine 43. 30. min.

CASE 3.

When the Radius is the second term.

Provided the third term be not greater than the first, transpose the terms. The me∣thod of transposition in this case is, as the first term is to the third, so is the second to the fourth, and then the work will be the same as in the second case. Ex. gr.

As the sine 30 is to the radius or sine 90, so is the sine 20 to what sine; which transpo∣sed is

As the sine 30 is to the sine 20, so is the radius to a fourth sine, which will be found 43, 30 min. as before.

CASE 4.

When the Radius is none of the three terms given.

In this case when both the middle terms are less than the first, enter the sine of the second term in the first, and laying the thread to the nearest distance, take the near∣est extent from the third to the thread: this distance applyed to the scale from the begin∣ning gives the fourth. Ex. gr.

As the sine 20 to the sine 10, so is the sine 30 to the sine 15.

When only the second term is greater than the first, transpose the terms and work as before.

But when both the middle tearms be great∣er than the first, this proportion will not be performed by this line without a paralel en∣trance or double radius; which inconveni∣ency shall be remedied in its proper place, when we shew how to work proportions by the lines of natural sines on the proportional or sector side.

These four cases comprizing the method of working all proportions by natural sines

alone, I shall propose some examples for the exercise of young practitioners, and there∣with conclude this Chapter.

PROBL. 4.

To finde the Suns amplitude in any Latitude.

As the cosine of the Latitude is to the sine of the Suns declination, so is the radius to the sine of amplitude.

PROBL. 5.

To finde the hour in any Latitude in Northern Declination.

Proport. 1. As the radius to the sine of the Suns declination, so is the sine of the latitude to the sine of the Suns altitude at six. By Probl. 2. substract this altitude at six from the present altitude, and take the difference. Then

Proport. 2. As the cosine of the latitude is to that difference, so is the radius to a fourth sine. Again

Proport. 3. As the cosine of the declina∣tion to that fourth sine, so is the radius to the sine of the hour from six.

PROBL. 6.

To finde the hour in any Latitude when the Sun is in the Equinoctial.

As the cosine of the latitude is to the sine of altitude, so is the radius to the sine of the hour from six.

PROBL. 7.

To finde the hour in any latitude in Southern De∣clination.

Proport. 1. As the radius to the sine of the Suns declination, so is the sine of the lati∣tude to the sine of the Suns depression at six; adde the sine of depression to the present al∣titude by Probl. 1. Then

Proport. 2. As the cosine of the latitude is to that summe, so is the radius to a fourth sine. Again,

Proport. 3. As the cosine of declination is to the fourth sine, so is the radius to the sine of the hour from six.

PROBL. 8.

To finde the Suns Azimuth in any latitude in Northern Declination.

Proport. 1. As the sine of the latitude to the sine of declination, so is the radius to the sine of altitude at East, or West. By Probl. 2. substract this from the present altitude, then,

Proport. 2. As the cosine of the latitude is to that residue, so is the radius to a fourth sine. Again,

Proport. 3. As the cosine of the altitude is to that fourth sine, so is the radius to the sine of the Azimuth from East or West.

PROBL. 9.

To finde the Azimuth for any latitude when the Sun is in the Equator.

Proport. 1. As the cosine of the latitude to the sine of altitude, so is the sine of the latitude to a fourth sine.

Proport. 2. As the cosine of altitude to that fourth sine, so is the radius to the sine of the Azimuth from East, or West.

PROBL. 10.

To finde the Azimuth for any latitude in South∣ern Declination.

Proport. 1. As the cosine of the latitude to the sine of altitude, so is the sine of the latitude to a fourth. Having by Probl. 4. found the Suns amplitude, adde it to this fourth sine by Probl. 1. and say

As the cosine of the altitude is to the sum, so is the radius to the sine of the Azimuth from East or West. The terms mentioned in the 5th. 7th. 8th. 10th. Problems are ap∣propriated unto us that live on the North side the Equator. In case they be applyed to such latitudes as lie on the South side the E∣quator. Then what is now called Northern declination, name Southern, and what is here styled Southern declination, term North∣ern, and all the proportion with the opera∣tion is the same.

These proportions to finde the hour and Azimuth, may be all readily wrought by the lines of artificial sines, only the addition and substraction must alwayes be wrought upon the line of natural sines.