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