M. Blundevile his exercises containing sixe treatises, the titles wherof are set down in the next printed page: which treatises are verie necessarie to be read and learned of all yoong gentlemen that haue not bene exercised in such disciplines, and yet are desirous to haue knowledge as well in cosmographie, astronomie, and geographie, as also in the arte of navigation ... To the furtherance of which arte of navigation, the said M. Blundevile speciallie wrote the said treatises and of meere good will doth dedicate the same to all the young gentlemen of this realme.

How to finde out the true distance of two places differing both in longitude and latitude by the Arithmeticall way. Chap. 14.

HOw is that done?

First, take the difference of the longi∣tudes and latitudes of both places by sub∣tracting the lesser out of the greater, then conuert the same into miles by multiply∣ing the difference of the two longitudes into the miles that be answerable to the latitude of each place, which miles you shall find in the Table aforesaid, and if there be any minutes annexed to the degrées of the difference of longitude, then reduce the same also to miles by the rule of pro∣portion, as before is taught, and hauing added the two products together, halfe the summe, and set it by it selfe. Then multiply the difference of the latitudes into 60. miles, and adde thereun∣to the fraction of minutes annexed to the said difference if it hath anie fraction, allowing for euerie minute one mile, and set that number also by it selfe: that done, square the summes reserued, that is to say, multiplie ech one part by it selfe into it selfe, and hauing added the two products together into one summe, séeke out the square roote thereof, & that shal be the true distance of the two places. As for example, if you would knowe the true distance betwixt London and Venice, first you must knowe by some Ta∣ble the longitude and latitude of both townes, wherefore finding the longitude of London to be 19. degrées, 54′· and the latitude thereof to be 51. degrées, 32′· And the longitude of Venice to be 35. degrées, 30′· and the latitude thereof to be 44. degrées, 45′· Now by subtracting the lesser longitude out of the greater, I finde the difference of longitude to bee 15. degrees, 36′· and by subtracting the lesser latitude out of the greater, I find the diffe∣rence of latitude to be 6. degrées, 47′· Then knowing the latitude of London to be 51. degrées, I resort to the table of miles ap∣pointed for euerie degrée of latitude before set downe, and theer I find that to 51. degrées of latitude do answere 37. miles and cer∣taine

seconds, which being of small moment are not woont to be reckoned. Then in multiplying the difference of the longitudes which is 15. degrées 36′· by 37. miles I find the product of the 15. degrées so multiplied to be 555. and because there bée 36′· annexed to the foresaid 15. degrées, I séeke by the rule of pro∣portion to knowe how many miles that fraction containeth, in saying thus, if 60. require 37. what shall 36. require? and I find 22. miles, which being added to 555. maketh 577. then by séeking in the foresaid table how manie miles be answerable to the lati∣tude of Venice, which is 44. degrées, I find the number of miles to be 43. by which number I multiply once againe the difference of longitude, which is 15. degrées. 36′· the product whereof, together with the fraction annexed thereunto being conuerted into miles by the rule of proportion, as before, doth a∣mount to 670. which sum being added to the former conuerted longitude, which is 577. maketh in all 1247. the halfe whereof is 623. which halfe number I reserue by it selfe, that done I multiply the difference of the latitude, which is 6. degrées 47′· by 60. miles, in saying 6. times 60. maketh 360. whereunto I adde for the 47′· annexed 47. miles, & it maketh in al 407. which sum I reserue also by it self. Then I multiplie the first reserued num∣ber into it selfe, the product whereof is 388129. That done, I multiplie the seconde reserued number also into it selfe, the product whereof is 165649. which two last productes being added together, do make in all 553778. whereof the square roote being taken, is 744. miles, which is the true distance of Venice by a right line from London. And to the intent that the order and working herein may more plainly appeare vnto you, I haue set downe all the particular numbers of the same here by them selues, as it were in a Table.

	longitude.	latitude
London.	19. degr. 54′·	51. degr. 32′·
Venice.	35. degr. 30′·	44. degr. 45′·
The difference of their longitudes and latitudes.	15. degr. 36′·	6. degr. 47′·

The difference of the longitudes conuerted into miles: for London is	577.
For Venice.	670.
The summe of the two conuerted longitudes ad∣ded together, is	1247.
The halfe whereof, which is the first reserued number, is,	623
The second reserued number, which is the diffe∣rence of the latitudes, conuerted into miles, is	407
The summe of the first reserued number mul∣tiplied into it selfe, is	388129
The summe of the second reserued number mul∣tiplied into itselfe, is	165649
The summe or both added together is	553778
The square root whereof, which is the summe of the miles, is	744

How to finde out the distance betwixt two places, diffe∣ring both in longitude and latitude by help only of a demi∣circle deuided into 180. degrees without any Arithmetike. Chap. 14.

BVt now because the way before taught to finde out such distance by the Arithme∣ticall way may séeme perhaps to some folkes very busie and tedious, I haue thought good therefore to set downe this other way which was sent me not long since from my louing friend M. Wright of Cayes colledge in Cambridge who is wel learned in the Mathematicals, & is so apt therunto by nature, as he is like inogh to attain to such perfit

knowledge therein as he may be able thereby hereafter greatly to perfect his country, if for want of sufficient exhibition he bée not forced to leaue so noble a studie, wherefore I wish with al my heart that all Gentlemen of abilitie were minded to shew their liberalitie towards him in that behalfe. But to returne to my matter, I say that the way to find out the foresaid distance is this here following: first hauing drawne a demi-circle vpon a right Diameter (the larger that the demi-circle is the better,) and di∣uided the same into 180. degrees, like vnto this hereafter descri∣bed, and marked with the letters a b c d. whereof d. is the cen∣ter and a. c. the Diameter. Then learne first by some Table to know the longitude the latitude of both places, and the diffe∣rence of their longitudes, as you did before in séeking to know by the Arithmetical way the distance betwixt London and Venice, the difference of whose two longitudes is 15. degrées, and 36′· as you may sée in the former Table: for in working by this way, you haue chiefly to séeke out in the circumference of the demicircle but thrée things, that is, first, the difference of the two longitudes, se∣condlie, the lesser latitude, and last of all the greatest latitude. Knowing therefore the difference of the said two places in longi∣tude to be 15. degrées, 36′· séeke out the same in the demicircle, beginning to count at A. and so proceed towards B. And at the end of those degrées and minutes set downe a pricke marked with the letter e. vnto which pricke drawe a right line by your ruler from D. the center of the demicircle. That done, séeke out the lesser latitude, which is 44. degrées, and 45′· in the forsaid demi∣circle, beginning to account the same from the pricke e. and so procéede towards the letter B. and at the end of the said lesser la∣titude, set downe another pricke marked with the letter g. from which prick or point draw a perpendicular line which by help of your squire or compasses may fal with right angles vpon the for∣mer right line drawne from D. to e. and where it falleth, there set downe a pricke marked with the letter h. That done, séeke out the greater latitude, which is 51. degrées, and 32′· in the foresaid demicircle, beginning to account the same from A. towards B. and at the end of that latitude set downe another prick marked with the letter I. from whence draw another perpendicular line that may fal by help of your squire or cōpasses with right angles

vpon the Diameter A. C. and there make a prick marked with the letter K. That done, take with your compasse the distance that is betwixt k. and h. which distance you must set downe vpon your said Diameter A. C. setting the one foote of your compasse vpon k. and the other towardes the center D. and there make a pricke marked with the letter L. Then take with your Compasse the length of the shorter perpendicular line g. h. and apply that widenesse vpon the longer perpendicular line I. K. setting the one foot of your Compasse at I. which is the end of the greater la∣titude, and extend the other foote towards K. and there make a pricke marked with the letter M. That done, take the distance betwixt L. and M. with your compasse, and apply the same to the demicircle, setting the one foote of your Compasse in A. and the other towards B. and there make a prick marked with the let∣ter N. And the number of degrées contained betwixt A. & N. will shew the true distance of the two places, which you shall find to be 12. degrées and almost 24′· Now by allowing for euery degrée 60. miles, and for euery minute a mile, the summe of miles will agrée with the former distance found out by the Arithmetical way which was 744. miles. And thus you haue to deale to know the distance of any other two places whatsoeuer, differing both in lon∣gitude and latitude. But you haue to note by the way, that if the difference of the longitudes doth excéede the number of 180. then you must subtract that excéeding difference out of 360. and the remainder shall be the difference of the longitudes, and then work in all points as is before taught. And this way is as Geome∣tricall, as that which Appian setteth downe in his booke of Ge∣ographie, to be done by the helpe of the terrestriall Globe, the or∣der whereof here followeth.

	Longitude.	latitude.
London.	19. deg. 54'.	52. deg. 32'.
Venice.	15. deg. 30'.	44. deg. 45'.
The difference of their longi∣tudes and lati∣tudes is	15. deg. 36'.	6. deg. 47'.

[illustration]