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. VI.

How to work proportions in Numbers, Sines, or Tangents, by the Artificial Lines thereof on the outward ledge.

THe general rule for all of these, is to extend the Compasses from the first term to the second (and observing whether that extent was upward or downward) with the same distance, set one point in the third term, and turning the other point the same way, as at first, it gives the fourth. But in Tangents when any of the terms exceeds 45, there may be excursions, which in their due place I shall remove.

PROBL. 1.

Numeration by the Line of Numbers.

The whole line is actually divided in∣to

100 proportional parts, and accordingly distinguished by figures, 1, 2, 3, 4, 5, 6, 7, 8, 9, and then, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100. So that for any number under 100, the Figures readily direct you, Ex. gr. To finde 79 on the line of numbers, count 9 of the small divisions beyond 70, and there is the point for that number. Now as the whole line is actually divided into 100 parts, so is every one of those parts subdivided (so far as conveniency will permit) actually into ten parts more, by which means you have the whole line actually divided into 1000 parts. For reckoning the Figures impressed, 1, 2, 3, 4, 5, 6, 7, 8, 9, to be 10, 20, 30, 40, 50, 60, 70, 80, 90, and the other figures which are stamped 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, to be 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000. You may enter any number under 1000 upon the line, accord∣ing to the former directions. And any num∣bers whose product surmount not 1000, may be wrought upon this line; but where the product exceeds 1000, this line will do no∣thing accurately: Wherefore I shall wil∣lingly omit many Problems mentioned by some Writers to be wrought by this line, as squaring, and cubing of numbers, &c. Sith they have only nicety, and nothing of exact∣ness in them.

PROBL. 2.

To multiply two numbers, given by the Line of Numbers. The proportion is this.

As 1 on the line is to the multiplicator, so is the multiplicand to the product. Ex. gr.

As 1 is to 4, so is 7, to what?

Extend the Compasses from the first term, viz. I unto the second term, viz. 4. with that distance, setting one point in 7 the third term, turn the other point of the Compasses toward the same end of the rule, as at first, and you have the fourth, viz. 28. There is only one difficulty remaining in this Problem, and that is to determin the number of places, or fi∣gures in the product, which may be resolved by this general rule. The product alwayes contains as many figures as are in the multi∣plicand, and multiplicator both, unless the two first figures of the product be greater than the two first figures in the multiplicator, and then the product must have one figure less than are in the multiplicator, and multi∣plicand both. Ex. gr. 47 multiplied by 25, is 2175, consisting of four figures; but 16 mul∣tiplied by 16, is 240, consisting of no more than three places, for the reason before men∣tioned.

I here (for distinction sake) call the multiplicator the lesser of the two numbers, although it may be either of them at plea∣sure.

PROBL. 3.

To work Division by the Line of Numbers.

As the divisor is to 1, so is the dividend to the quotient.

Suppose 800 to be divided by 20, the quo∣tient is 40. For,

As 20 is to 1, so is 800 to 40.

To know how many figures you shall have in the quotient, take this rule. Note the dif∣ference of the numbers of places or figures in the dividend and divisor. Then in case the quantity of the two first figures to the left hand in your divisor be less than the quantity of the two first figures to the left hand in your dividend, the quotient shall have one figure more than the number of difference: But where the quantity of the two first figures of the divisor is greater than the quantity of the two first figures of the dividend, the quotient will have only that number of figures noted by the difference. Ex. gr. 245 divided by 15. will have two figures in the quotient;

but 16 divided 〈…〉〈…〉 ••••ve only one fi∣gure in the quotient.

PROBL. 4.

To finde a mean proportional 'twixt two Numbers given by the line of Numbers.

Divide the space betwixt them upon the line of numbers into two equal parts, and the middle point is the mean proportional. Ex. gr. betwixt 4 and 16, the mean proportional is 8. If you were to finde two mean propor∣tionals, divide the space 'twixt the given num∣bers into three parts. If four mean propor∣tionals divide it into five parts, and the seve∣ral points 'twixt the two given numbers, will show the respective mean proportionals.

PROBL. 5.

To work proportions in Sines alone, by the Arti∣ficial Line of Sines.

Extend the Compasses from the first term to the second, with that distance set one point in the third term, and the other point gives the fourth. Only observe that if the second term be less than the first, the fourth

must be less 〈…〉〈…〉 or if the second term exceed 〈…〉〈…〉 fourth will be greater than the third. This may direct you in all proportions of sines and tangents singly or conjunctly, to which end of the rule to turn the point of your Compasses, for finding the fourth term. Ex. gr.

As the sine 60 is to the sine 40, so is the sine 20 to the sine 14. 40. Again,

As the sine 10 is to the sine 20, so is the sine 30 to the sine 80.

PROBL. 6.

To work proportions in Tangents alone by the Ar∣tificial Line of Tangents.

For this purpose the artificial line of tan∣gents must be imagined twice the length of the rules, and therefore for the greater con∣veniency, it is doubly numbred, viz. First from 1 to 45, which is the radius, or equal to the sine 90: In which account every di∣vision hath (as to its length on the rule) a proportional decrease. Secondly, its num∣bred back again from 45 to 89, in which ac∣count every division hath (as to its length on the line) a proportional encrease. So that the tangent 60 you must imagine the whole

length of the Rule; and so much more as the distance from 45 unto 30 or 60 is. This well observed, all proportions in tangents are wrought after the same manner of extend∣ing the Compasses from the first term to the second, and that distance set in the third, gives the fourth, as was for sines and numbers. But for the remedying of excursions, sith the line is no more than half the length, we must imagine it. I shall lay down these Cases.

CASE 1.

When the fourth term is a tangent exceeding 45, or the Radius. Ex. gr.

As the tangent 10 is to the tangent 30, so is the tangent 20, to what?

Extending the Compasses from 10 on the line of tangents to 30, with that distance I set one point in 20, and finde the other point reach beyond 45, which tells me the fourth term exceeds 45, or the radius; wherefore with the former extent, I set one point in 45; and turning the other toward the be∣ginning of the line, I mark where it touch∣eth, and from thence taking the distance to the third term, I have the excess of the fourth

term above 45 in my compass: wherefore with this last distance setting one point in 45, I turn the other upon the line, and it reach∣eth to 50, the tangent sought.

CASE 2.

When the first term is a tangent exceeding 45, or the Radius. Ex. gr.

As the tangent 50 is to the tangent 20, so is the tangent 30, to what?

Because the second term is less than the first, I know the fourth must be less than the third. All the difficulty is to get the true ex∣tent from the tangent 50 to 20. To do this, take the distance from 45 to 50, and setting one point in 20, the second term, turn the other toward the beginning of the line, mark∣ing where it toucheth, extend the Compasses from the point where it toucheth to 45, and you will have the same distance in your Com∣passes as from 50 to 20, if the line had been continued at length unto 89 tangents, with this distance, set one point in 30 the third term, and turn the other toward the begin∣ning (because you know the fourth must be less) and it gives 10 the tangent sought.

CASE. 3.

When the third term is a Tangent exceeding 45, or the Radius.

As the tangent 40 is to the tangent 12, 40 min. so is the tangent 65, to what?

Extend the Compasses from 40 to 12 d. 40 min. with distance, setting one point in 65. turn the other toward 45, and you will finde it reach beyond it, which assures you the fourth term will be less than 45. Therefore lay the extent from 45 toward the beginning, and mark where it toucheth, take the distance from that point to 65, and laying that distance from 45 toward the be∣ginning it gives 30, the tangent sought.

These Cases are sufficient to remove all difficulties. For when the second term ex∣ceeds the Radius, you may transpose them, saying, as the first term is to the third; so is the second to the fourth, and then its wrought by the third Case.

I suppose it needless to adde any thing about working proportions by sines and tan∣gents conjunctly, sith, enough hath been al∣ready said of both of them apart, in these two last Problems; and the work is the same

when they are intermixed. Only some pro∣portions I shall adjoyn, and leave to the pra∣ctice of the young beginner, with the di∣rections in the former Cases.

PROBL. 7.

To finde the Suns ascensional difference in any Latitude.

As the co-tangent of the latitude is to the tangent of the Suns declination, so is the ra∣dius to the sine of the ascensional difference.

PROBL. 8.

To finde at what hour the Sun will be East, or West in any Latitude.

As the tangent of the latitude is to the tangent of the Suns declination, so is the ra∣dius to the cosine of the hour from noon.

PROBL. 9.

The Latitude, Declination of the Sun, and his Azimuth from South, given to finde the Suns Altitude at that Azimuth.

As the radius to the cosine of the Azimuth from south, so is the co-tangent of the lati∣tude,

to the tangent of the Suns altitude in the equator at the Azimuth given. Again,

As the sine of the latitude is to the sine of the Suns declination, so is the cosine of the Suns altitude in the equator (at the same Azi∣muth from East or West) to a fourth ark.

When the Azimuth is under 90, and the latitude and declination is under the same pole, adde this fourth ark to the altitude in the equator. In Azimuths exceeding 90, when the latitude and declination is under the same pole, take the equator altitude out of the fourth ark. Lastly, when the latitude and declination respect different poles, take the fourth ark out of the equator altitude, and you have the altitude sought.

PROBL. 10.

The Azimuth, Altitude, and Declination of the Sun, given to finde the hour.

As the cosine of declination is to the sine of the Suns Azimuth, so is the cosine of the altitude to the fine of the hour from the Me∣ridian.

Proportions may be varied eight several wayes in this manner following.

• 1. As the first term is to the second, so is the third to the fourth.
• ...

• 2. As the second term is to the first, so is the fourth to the third.
• 3. As the third term is to the first, so is the fourth to the second.
• 4. As the fourth term is to the second, so is the third to the first.
• 5. As the second term is to the fourth, so is the first to the third.
• 6. As the first term is to the third, so is the second to the fourth.
• 7. As the third term is to the fourth, so is the first to the second.
• 8. As the fourth term is to the third, so is the second to the first.

By thesse any one may vary the former pro∣portions, and make the Problems three times the number here inserted. Ex. gr. To finde the ascensional difference in Problem 10, of this Chapter, which runs thus.

As the co-tangent of the latitude is to the tangent of the Suns declination, so is the radius to the sine of ascensional difference. Then by the third variety you may make an∣other Problem, viz.

As the radius is to the co-tangent of the latitude, so is the sine of the Suns ascensional difference to the sine of his declination. A∣gain, by the fourth variety you may make a third Problem, thus,

As the sine of the Suns ascensional diffe∣rence is to the tangent of the Suns declinati∣on, so is the radius to the co-tangent of the latitude.

By this Artifice many have stuffed their Books with bundles of Problems.