Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

PROP. VII. By knowing the Latitudes of two places, and like∣wise their Difference of Longitude; to find,

• 1. The distance in the Arch.
• 2. The direct Position from the first to the second place.
• 3. The direct Position from the second to the first place.
• 4. The Latitudes and Longitudes by which the Arch passeth.
• 5. The Course and Distance from Place to Place through those Latitudes and Longitudes according to Mercator.

I shall here make use of M. Norwood's example of a Voyage from the Summer-Islands, unto the

Lizard: now because the work is various I have therefore illustrated it with a Scheme, and shall be as brief and facile as possible. Therefore,

In the Triangle ADE, let A be the Summer∣Islands, whose Latitude is 32° 25', AD the Com∣plement* 1.1 thereof 57° 35', let E represent the Lizard whose Latitude is 50° 00', and ED the Complement thereof 40° 00', and let their Dif∣ference of Longitude, namely the Angle ADE be 70° 00', now Drepresenteth the North-Pole, and AE an Arch of a great Circle passing by these two Places: now see the operation.

1. By having the Complements of the Latitudes of the two Places, viz. AD 57° 35', and ED 40° 00', and their Difference of Longi∣tude, namely the Angle EDA 70° 00': you may find the nearest distance EA to be 53° 24'; by Case the 9. § 5. chap. 5.

2. Then having found the nearest distance in* 1.2 the Arch EA to be 53° 24', (or 3204 Miles) the Angle of Position from the Summer Islands to the Lizard, namely the Angle DAE, may be found by Case the 1. § 5. chap. 5. to be 48° 48', that is N. E. and 03° 48' Easterly.

3. And also by the same Case, may the Direct Position from the Lizard, to the Summer-Islands, namely the Angle AED befound to be 81° 10', that is W. by N. and 2° 25' Westerly.

4. In order to the finding the Latitudes and Longitudes by which the Arch passeth, first let fall the Perpendicular DB, so is the Oblique* 1.3 Triangle ADE converted into two Rectangulars, viz. ABD, and DBE: secondly, by Case the 8. § 4. chap. 5. you may find the length of the

Perpendicular DB to be 39° 26', whose Comple∣ment is 50° 34', which is the greatest Latitude by which the Arch ABE passeth, so the greatest Ob∣liquity* 1.4

BDc	48°	31
BDf	38	31
BDg	28	31
BDh	18	31
BDj	08	31

of the Equinoctial from that Circle is 50° 34'. — Thirdly, by Case the 9. § 4 chap. 5. you must find the vertical Angles, viz. ADB, and BDE, which will appear, the Angle ADB to be 58° 31', and EDB to be 11° 29': now these things being obtained, the Lati∣tudes by which the Arch passeth at every tenth degree of Longitude from A, may be found by resolving the several Right-Angled Triangles, viz. BDc, BDf, &c. substracting 10° from ADB 58° 31', there remains BDc 48° 31', and so for the rest as in the Table. Now by knowing these Angles last found, and the Perpendicular BD before found to be 39° 26', you may by Case the 3. §. 4. chap. 5. find the Latitudes of the several points A. c. f. g. h. i. B. and E. to be as in the subsequent Table.

5. Thus having* 1.5

Latitude.	Longitude.
A.	32°	25'	00	00'
c.	38	51	10	00
f.	43	34	20	00
g.	46	54	30	00
h.	49	04	40	00
i.	50	15	50	00
B.	50	34	60	00
E.	50	00	70	00

found the Latitudes and Longitudes of the Arch, and the other required parts afore∣mentioned, we now come to shew how the Course, and the Distance from place to place according to Mercator may be found. So to find, first the Course and Distance

Ac. now there is given the Latitude of A 32° 25', and of c 38° 51', and their Difference of Longitude is 10° 00', now the Proper Difference* 1.6 of Latitude is 6° 26', or 386', and Meridional Difference of Latitude is 475'. Now knowing these things by proposition 2. § 2. chap. 8. yon may find the Course from A to c, to be N. E. 51° 38'; and the Distance Ac to be 622', and so those Rules prosecuted will shew the course and distance from c to f; from f to g; from g to h, &c. So of the rest, which for brevity sake I shall omit, and leave the Ingenious Seaman to Calculate at his Pleasure.

I might hereunto annex many more proposi∣tions of Circular Sailing, but because of the smallness of this Treatise, and that those Pro∣positions already handled, being by the Inge∣nious Seaman well understood, will be sufficient to enable him to perform any other Conclusion in Circular Sailing whatsoever, I therefore here omit, and hasten forwards unto the other parts of this Mathematical Treasury.

A Table of Angles, which every Rumb makethwith the Meridian.

These on this side the W. incline to∣wards the N. end of the Meridian	Angles of In∣clination with the Meridian.	These on this side the E. incline to the N. end of the Meridian.
Rumbs.	North	Rumbs.
N. by W.	11°	15'	N. by E.
N. N. W.	22	30	N. N. E.
N. W. by N	33	45	N. E. by N.
North West	45	00	North East
N. W. by W.	56	15	N. E. by E.
W. N. W.	67	30	E. N. E.
W. by N.	78	45	E. by N.
West	90	00	East
W. by S.	78	45	E. by S.
W. S. W.	67	30	E. S. E.
S. W. by W.	56	15	S. E. by E.
South West	45	00	South East
S. W. by S.	33	45	S. E. by S.
S. S. W.	22	30	S. S. E.
S. and by W.	11	15	S. and by E.
Rumbs	South	Rumbs
These on this side the W. in∣cline unto the S. end of the Meridian.	These on this side the E. incline to∣wards the S. end of the Meridian.
Note that if you account in quarter of Points, add for one quarter 2° 48', for one half 5° 37', for three quarters 8° 26', (not regarding the Seconds in Navigation.)

Notes

• * 1.1

  Fig. 51.
• * 1.2

  Fig. 51.
• * 1.3

  Fig. 51.
• * 1.4

  Fig. 51.
• * 1.5

  Fig. 51.
• * 1.6

  Fig. 51.