Gunter's Books of the Sector

 The First Book of the Sector 

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CHAP. I. 
The Description, the Making , and the General Use of the SECTOR. 
1 
15 

1. The Description of the SECTOR 
1 
15 

2. Of the making of the SECTOR 
3 
16 

3. To divide the Lines of Superficies 
4 
17 

4. To divide the Lines of Solids. 
6 
18 

5. To divide the Lines of Sines and Tangents on the Sides of the Sector 
11 
20 

6. To shew the Ground of the Sector. 
12 
21 

7. To shew the General Use of the Sector. 
13 
21 
CHAP. II. 
The Use of the Scale of Lines. 
17 
23 

1. To set down a Line, resembling any given parts or Fraction of Parts. 
17 
23 

2. To increase a Line in a given Proportion. 
18 
24 

3. To diminish a Line in a given Proportion. 
18 
24 

4. To divide a Line into any number of Parts given. 
18 
24 

5. To find a Proportion between two or more right Lines given. 
19 
24 

6. Two Lines being given, to find a third in continual Proportion. 
20 
24 

7. Three Lines being given, to find a fourth in discontinual Proportion. 
21 
24 

8. To divide a Line in such sort as another Line before is divided. 
22 
25 

9. Two Numbers being given, to find a third in continuous Proportion. 
23 
25 

10. Three Numbers being given, to find a fourth in discontinual Proportion. 
24 
27 
CHAP. III. 
The Use of the Lines of Superficies. 
26 
28 

1. To find a Proportional between two or more like Superficies. 
26 
28 

2. To augment a Superficies in a given Proportion. 
26 
28 

3. To diminish a Superficies in a given Proportion. 
26 
28 

4. To add one like Superficies to another. 
27 
28 

5. To subtract one like Superficies from another. 
27 
28 

6. To find a mean Proportional between two Lines given. 
28 
29 

To make a Square equal to a Superficies given. 
28 
29 

To find a Proportion between Superficies, though they be unlike one to another. 
29 
29 

To make a Superficies, like to one Superficies, and equal to another. 
29 
29 

7. To find a mean Proportion between two Numbers given. 
30 
30 

8. To find the Square Root of a Number. 
31 
30 

9. The Root being given, to find the Square Number of that Root. 
31 
30 

10. Three Numbers give, to find a fourth in a duplicate Proportion. 
32 
31 
☞ 
11. How to describe a Parabola, by help of the Line of Lines and Superficies. 
35 
32 
CHAP. IV. 
The Use of the Line of Solids. 
36 
33 

1. To find a Proportion between two or more like Solids. 
36 
33 

2. To augment a Solid in a given Proportion 
37 
33 

3. To diminish a Solid in a given Proportion 
37 
33 

4. To add one like Solid to another. 
37 
33 

5. To subtract one like Solid from another. 
37 
33 

6. To find two mean proportionals Lines between two extreme Lines given. 
38 
34 

7. To find two mean proportionals Numbers between two extreme Numbers given. 
39 
34 

8. To find the Cubic Root of a Number. 
40 
35 

9. The Root being given, to find the Cube Number of that Root. 
40 
35 

10. Three Numbers given, to find a fourth in a triplicate Proportion. 
41 
35 

The Second Book of the Sector. Containing the Use of the Circular Lines. 
CHAP. I. 
Of the nature of Sines, Chords, Tangents, and Secants, fit to be known beforehand, in reference to rightlined Triangles. 
43 
36 
CHAP. II. 
Of the General Use of Sines and Tangents. 
48 
39 

1. The Radius being known, to find the Sine of any Ark or Angle. 
48 
39 

2. The right Sine of any Ark being given to find the Radius 
49 
39 

3. The Radius of a Circle, or the right Sine of any Ark, being given, to find a streight Line resembilng a Sine, to find the quantity of that Sine. 
47 
39 

4. The Radius or any right Sine being given, to find the Versed Sine of any Ark. 
48 
40 

5. The Diameter or Radius being given, to find the Chords of every Ark. 
48 
40 

Having two right Lines resembling the Chord and Versed Sine, to find the Radius and Diameter. 
50 
41 

6. The Chord of any Ark being given, to find the Diameter and Radius. 
51 
41 

Having the Diameter of an Ellipsis to describe the same upon a plane. 
52 
42 

7. To open the Sector to any quantity of any Angle given. 
54 
44 

8. The Sector being opened, to find the quantity of the Angle. 
54 
44 

9. To find the quantity of any Angle given. 
55 
44 

10. Upon a right Line, and a Point given in it, to make an Angle equal to any Angle given. 
56 
45 

11. To divide the Circumference of a Circle given into any parts required. 
56 
45 

12. To divide a right Line by extreme and mean proportion. 
57 
45 
CHAP. III. 
Of the Projection of the Sphere in Plano. 

47 

SECT. I. To Project a Sphere in Plano, by streight Lines. 

47 

Some Uses of this Projection. 
60 
48 

SECT. II. To Project a Sphere in Plano, by circular Lines. 
61 
48 

Some Uses of this Projection. 
63 
49 

The Use of this Nocturnal. 
65 
51 

SECT. III. Another way to Project the Sphere by Circular Lines. 
65 
51 

Some Uses of this Projection. 
67 
52 

SECT. IV. A Third way to Project the Sphere in Plano, by Circular Lines. 
68 
53 

Some Uses of this Projection. 
70 
56 
☞ 
1. For a Horizontal Dial 
71 
56 
☞ 
2. For an Erect direct North or South Dial. 
71 
56 
☞ 
3. For a Vertical Declining Dial. 
71 
56 
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4. For direct Incliner. 
72 
57 
☞ 
5. For Declining inclining Planes. 
72 
57 
☞ 
SECT. V. Of the Projection of the Sphere upon Oblique Circles. 
72 
57 
CHAP. IV. 
Of the Resolution of rightlined Triangles. 
76 
59 

1. In a Rectangled Triangle, to find the Base, both sides being given. 
76 
59 

2. To find the Base, by having the Angles, and one of the sides given. 
77 
59 

3. To find a side, by having the Base and the other side given. 
78 
60 

4. To find a side by having the Base, and the Angles give. 
78 
60 

5. To find a side, by having the other side, and the Angles given. 
79 
60 

6. To find the Angles, by having the Base and one of the sides given. 
79 
60 

7. To find the Angles, by having both sides given. 
79 
60 

8. The Radius give, to find the Tangent and Secant of any Ark. 
79 
60 

9. The Tangent of any Ark being given, to find the Secant thereof, and the Radius. 
80 
61 

10. The Secant of any Ark being given, to find the Tangent, and the Radius. 
80 
61 

11. In any rightlined Triangle whatsoever, to find a side by knowing the other two sides, and the Angle contained by them. 
80 
61 

12. To find a side by having the other two sides, and one of the adjecent Angles, so it be known which of the other Angles is Acute or Oblique. 
81 
61 

13. To find a side by having the Angles, and one of the other sides given. 
81 
61 

14. To find the Proportion of the sides, by having the three Angles. 
82 
62 

15. To find an Angles, by having the three sides. 
82 
62 

16. To find an Angle, by having two sides, and one adjecent Angle. 
83 
62 

17. To find an Angle, by having two sides, and the Angle contained by them. 
83 
62 
CHAP. V. 
Of the Resolution of the Sperical Triangle. 
85 
63 

1. In a Rectangled Triangle, to find a side, by knowing the Base, and the Angle opposite to the reqiured side. 
87 
64 

2. To find the side, by knowing the Base, and the other side. 
87 
64 

3. To find a side, by knowing the two Oblique Angles. 
88 
65 

4. To find the Base, by knowing both the sides. 
88 
65 

5. To find the Base, by knowing one side, and the Angle opposite to that side. 
88 
65 

6. To find an Angle, by the other Oblique Angels, and the side opposite to the inquired side. 
89 
65 

7. To find an Angle, by the other two Oblique Angles, and the side opposite to that Angle given. 
89 
65 

8. To find an Angle, by the Base, and the side, opposite to that inquired Angle. 
89 
65 

9. To find a side, by having the other side, and the Angle opposite to the inquired side. 
90 
66 

10. To find a side, by having the other side, and the Angle next to the inquired side. 
90 
66 

11. To find a side, by having the Base, and the Angle adjacent next to the inquired side. 
91 
66 

12. To find the Base, by knowing the Oblique Angles. 
91 
66 

13. To find the Base, by one of the sides, and the Angle adjacent next that side. 
91 
66 

14. To find an Angle, by knowing both the sides. 
92 
67 

15. To find the Angle, by the Base, and the side adjacent next to the required Angle. 
92 
67 

16. To find an Angle, by knowing the other Oblique Angle, and the Base. 
93 
67 

17. In any Spherical Triangle whatsoever, to find a side, opposite to an Angle given, by knowing one side and two Angles, whereof one is oppisite to the given side, the other to the side required. 
93 
67 

18. To find an Angle opposite to a side given, by having one Angle and two sides, the one opposite to the given Angle, the other to the Angle required. 
94 
68 

19. To find an Angle, knowing the three sides. 
94 
68 

20. To find a side, by knowing the three Angles. 
96 
69 

21. To find a side, by having the other two sides, and the Angle comprehended. 
96 
69 

22. To find a side, by having the other two sides, and one of the Angles next the inquired side. 
97 
69 

23. To find a side, by having one side, and the two Angles next to the inquired sides. 
97 
69 

24. To find a side, by having two Angles, and the side inclosed by them. 
98 
70 

25. To find an Angle, by having the other two Angles, and the side inclosed by them. 
98 
70 

26. To find a side, by having the other two Angles, and one of the sides next to the inquired Angle. 
98 
70 

27. To find an Angle, by knowing two sides, and the Angle contained by them. 
98 
70 

28. To find an Angle, by knowing the two sides next to it, and one of the other Angles. 
99 
70 
CHAP. VI. 
Of the Use of the Meridian Line in Navigation. 
99 
70 

1. To divide a SeaChart according to Mercator's Projection. 
99 
70 

A Table for the Division of the Meridian Line. 
100 
71 


107 
76 

2. To find how many Leagues answer to one Degree of Longitude in every several Lattitude. 
111 
78 

3. To find how many Leagues do answer to one Degree of Lattitude in every several Rumb. 
113 
80 

1. By one Latitude, Rumb, and Distance, to find the difference in Latitude. 
115 
81 

2. By the Rumb and both Latitudes, to find the Diastance upon the Rumb. 
116 
82 

A Table of Leagues, Rumb, and Difference of Latitude. 
117 
82 

3. By the Distance, and both Latitudes, to find the Rumb. 
120 
84 

4. By the Longitude and Latitude of both places, to find the Rumb. 
121 
84 

5. By the Rumb and both Latitudes, to find the difference of Longitude. 
122 
85 

Table 
124 
87 


131 
92 

6. By the difference of Longitude, Rumb and one Latitude, to find the other Latitude. 
133 
93 

7. By one Latitude, Rumb, and the Distance, to find the difference of Longitude. 
134 
95 

8. By one Latitude, Rumb, and difference of Longitude, to find the distance. 
135 
95 

9. By one Latitude, distance and difference of Longitude, to find the Rumb. 
136 
96 

10. By the Longitude, and Latitude of two places, to find their diatnce upon the Rumb. 
137 
96 

11. By the Latitude of two places, and the distance upon the Rumb, to find the difference of Longitude. 
138 
97 

12. By one Latitude, distance, and difference of Longitude, to find the difference of Latitudes. 
139 
97 

The Third Book of the Sector. Containing the Use of the particular Lines. 
CHAP. I. 
Of the Line of Quadrature. 
141 
98 

1. To make a Square equal to a Circle given. 
141 
98 

2. To make a Circle equal to a Square given. 
141 
98 

3. To reduce a Circle give, or a Square, into an euqal Pentagon, or other like sided and like angled figure. 
142 
99 

4. To find a right line, equal to the Circumference of a Circle, or other part thereof. 
143 
99 
CHAP. II. 
Of the Lines of Segments. 
143 
99 

1. To divide a Circle given into two Segments according to a Proportion given. 
143 
99 

2. To find a Proportion between a Circle and his Segments given. 
144 
102 

3. To find the side of a Square, equal to any known Segment of a Circle. 
145 
102 
CHAP. III. 
Of the Lines of Inscribed Bodies. 
145 
102 

1. The Semidiameter of a Sphere being given, to find the sides of the five regular Bodies, which may be inscribed in the said Sphere. 
145 
102 

2. The side of any of the five regular Bodies being given, to find the Semidiameter of a Sphere, that will circumscribe the said Body. 
145 
102 
CHAP. IV. 
Of the Lines of Equated bodies. 

104 

1. The Diameter of a Sphere being given, to find the side of the five regular Bodies, equal to the Sphere. 
146 
104 

2. The side of any of the five regular Bodies being given, to find the Diameter of a Sphere, and the sides of the other bodies, equal to the first body given. 
146 
104 
CHAP. V. 
Of the Lines of Metals. 

104 

1. In like bodies of several Metals, and equal weight, having the magnitude of the one, to find the magnitude of the rest. 
147 
104 

2. In like bodies of several Metals, and equal magnitude, having the weight of one, to find the weight of the rest. 
147 
104 

3. A body being given of one Metal, to make another like unto it, of another Metal and equal weight. 
148 
105 

4. A body being given of one Metal, to make another like unto it, of another Metal, according to a weight given. 
148 
105 
CHAP. VI. 
Of the Lines on the edges of the Sector. 

106 

To observe the Altitude of the Sun. 
150 
106 

1. To draw the hour Lines upon a Horizontal Plane. 
150 
106 

2. To draw the hour Lines upon a direct Vertical Plane. 
150 
106 

3. To draw the hour Lines on a Polar Plane. 
152 
107 

4. To draw the hour Lines on a Meridian Plane. 
152 
107 

5. To draw the hour Lines on a Vertical Declining Plane. 
152 
107 

6. To prick down the hour points another way. 
154 
108 

The Conclusion to the Reader
Gresham Coll. 1 Maij 1623, E.G. 
156 
109 

FINIS 



☞ 
The Sector Altered; and other Scales added: with the Description and Use thereof.
Invented and Written by Mr. Samuel Foster, sometimes publick Professor of Astronomy in Gresham College in London. (Printed 1673). 
CHAP. I. 
Of the Sector in General. 
159 
110 
CHAP. II. 
How the several Lines are disposed upon the Sector. 
160 
111 

Of the other Lines inscribed on the edges and spare places of the Sector. 
161 
111 
CHAP. III. 
The Genral Use of the Sector, and the manner of working on it. 
162 
112 

How by three terms given in any kind, how to find a fourth. 
162 
112 

The manner of working them in general, according to these two rules, will be this: 
163 
112 
CHAP. IV. 
Examples of several kinds. 
166 
114 

1. Three numbers 52, 39, 44, being given, to find a fourth proportional. 
166 
114 

2. Three Sines being given, to find a fourth proportional Sine. 
167 
114 

3. As the Sine of 60 gr. is to the number 35, so the Sine of 48 gr. to what number? 
167 
114 

4. As the Sine of 60, is to the Tangent of 55 gr. So the Sine of 50 to the tangent of what Ark? 
168 
115 

5. Having three numbers, to find a fourth in duplicate proportion. 
169 
115 

6. Having two numbers, to find a mean proprtional. 
170 
116 

7. Having three numbers given, whereof the two first are supposed to be in a duplicate proportion, how to find a fourth, unto which the third shall be in the simple proportion of the former; that is, As the square root of the first to the square root of the second. 
170 
116 

8. Having three numbers, to find a fourth in a triplicate proportion. 
171 
116 

9. How to find two mean proportionals between two numbers given. 
171 
116 

10. Having three numbers given, whereof the two first are supposed to be in a triplicate proportion, how to find a fourth, unto which the third shall be in the simple proportion of the former; that is, As the cubic root of the first to the cubic root of the second. 
172 
117 
CHAP. V. 
Of the Scale of Chords. 
172 
117 

1. The Line of Chords is numbered up to 90, and will therefore set of or measure any Ark within 90 gr. But if the Ark be more, it must do it at twice or more times. 
173 
117 

2. Having the Radius DA, and the Cord AB assigned, I would know to what number of degrees that Cord answereth. 
174 
118 

3. Let AB be the Cord of 79 gr. given, and the radius to which it is estimated to be such a Chrod required. 
174 
118 

(inserted Table: A brief Synopsis of this Oblique Projection.) 


CHAP. VI. 
Of the Tangents and Secants. 
175 
121 

1. Having any line given, as a known tangent or Secant: To find the Radius belonging to it. 
175 
121 

2. Having the Radius, to find any Tangent or Secant belonging thereto. 
176 
122 

Corollary 
176 
122 
CHAP. VII. 
How to supply the Meridian Line or Line of Rumbs, by the Scale of Secants. 
176 
122 

1. How to make a Seachart, after Mercators Projection. 
176 
122 

2. The uses of the SeaChart, and some other Propositions that concern Navigation, are set down by Mr. Gunter lib. 2, chap. 6 of this Sector, which may here also be done. 
177 
122 
CHAP. VIII. 
The uses of the Line of Versed Sines.. 
177 
122 

1. Having two sides of a Spherical Triangle, and the Angle comprehended, to find the Base. 
178 
123 

2. Having the three sides of a Spherical Triangle to find the Vertical Angle. 
179 
123 

3. Having a proportion to be wrought in Sine alone, wherein the radius leads in the proportion, how to find a fourth proportional Sine upon these Versed Sines. 
179 
123 

To get the Suns Azimuth. 
180 
123 

To find the hour of the day. 
180 
123 

To find the Suns Altitude at any hour. 
180 
123 

Corollary 
181 
124 

To find the Amplitude of the raising and setting of the Sun. 
181 
124 

The Latitude of two places, and their distance being given, to find their difference in Longitide. 
182 
125 

Having the Latitude, the Suns place in the Zodiac, and the Altidute above the Horizon, to find the hour from the South. 
182 
125 

1. Corollary To find the Semidiurnal and Seminocturnal Ark. 
183 
125 

2. Corollary To find the moment of time, when the Crepuiculum begins and ends. 
184 
126 

3. Corollary The Suns place being assigned in any place of the Zodiac, to find the Altitude at all hours. 
184 
126 
CHAP. IX. 
(number missing) 


CHAP. X. 
Of the other Scales on the edges and other spare places of the Sector. 
188 
128 
CHAP. XI. 
Of the two Scales of wine and ale measure. 
188 
128 
CHAP. XII. 
How to perform the same work of Gauging by Feet or Inches. 
190 
129 

To find the Content in Galons. 
191 
129 
CHAP. XIII. 
Of the two Scales upon the inner edge of the other Leg, which are divided the one unto 14, the other unto 20 equal parts. 
192 
130 

To Gauge by the mean Diameter. 
192 
130 
CHAP. XIV. 
How to measure Cartridges of Gunpowder to know howmany pound are contained in them. 
193 
130 

1. If the Cartridge be of a cylindrical form. 
193 
130 

2. If the Cartridge be of a conical form. 
193 
130 

3. If the Cartridge be a resected cone. 
194 
131 

Postscript 
195 
131 