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.

The definitions of the foresaid tearmes.

AN Arch is any part or portion of the circumference of a Cir∣cle, which in this practise doth not commonly extend beyond 180. degrées which is one halfe of the circumference of any Circle how great or small so euer it be, for euery Circle containeth 360 degrées.

A Chord is a right line drawne from one end of the Arch to the other end thereof, and note that all chordes are alwaies lesser then the Diameter it selfe, for that is the greatest chord in any Circle.

Sinus rectus is the one halfe of a Chord or string of any Arke

which is double to the Arke that is geuen or supposed, and falleth with right Angles vpon y^t Semidiameter which diuideth the dou∣ble Arke into two equall parts.

Sinus versus that is to say turned the contrary way, is a right line, and that part of the Semidiamiter, which is intercepted be∣twixt the beginning of the giuen Arke and the right Sine of the same Arke, and this is also called in Latine Sagitta, in English a Shaft or Arrowe, for the Demonstratiue figure thereof hereafter following, is not vnlike to the string of a bowe ready bent hauing a Shaft in the midst thereof.

Quadrans is the fourth part of a Circle containing 90. de∣grées.

Complementum arcus, is that portion of the Circle, which sheweth how much the giuen Arke is lesser then the quadrant, if the giuen Arke doe containe fewer degrées then the Quadrant, but if it containe more degrées then the Quadrant, then the diffe∣rence betwixt such giuen arch, and the halfe Circle is the comple∣ment of the said giuen Arke.

Sinus complementi, is the right Sine of that Arch which is the complement of the giuen Arke.

Sinus totus, is the Semidiamiter of the Circle, & is the grea∣test Sine that may be in the Quadrant of a Circle, which accor∣ding to the first tables of Monte Regio containeth 6/000/000. and according to the last tables 10/000/000 parts, for the more parts that the totall Sine hath, the more true and exact shall your worke be, notwithstanding sometime it shall suffice to attribute vnto the totall Sine but 60/000. parts, which number Appian obserueth in teaching the way to find out the distance of two pla∣ces differing both in Longitude and Latitude by the tables of Sines, and some doe make the totall Sine to containe 100/000 parts, as Wittikindus in his treatise of Dials, and diuers o∣ther doe the like. Also Clauius himselfe saith that in the tables set downe by him in quarto, you may sometime make the totall Sine to be but 100/000. so as you cut off the two last figures on the right hand in euery Sine, but you shall better vnderstand euery thing here aboue mentioned, by the figure Demonstratiue here following.

[illustration]

The figure Demonstratiue.

In this figure you sée first a whol Circle drawne vp∣on the Centre E. and marked with the letters A. B. C D. which Circle by two crosse Di∣ameters marked with the letters A C. and B. D. & pas∣sing both through the Centre E. is di∣uided into foure Quadrantes or quarters, the vp∣per Quadrante whereof on the left hand is marked with the letters A. B. E. in which Quadrant, the right perpendicular line marked with the letters F. H. betokeneth the right Sine of the giuen Arke A. F. which right Sine is the one halfe of the chord or string F. G. and the giuen Arke A. F. is the one halfe of the double Arke or bowe G. A. F. and A. H. is the Shaft called in Latine Sinus versus: Againe the letters F. B. doe shew the complement which together with the giuen Arke A. F. doe make the whole Quadrant A. F. B which is diuided into 9. spaces, euery space cōtaining 10. degrées, whereby you may plainely perceiue that in this demonstration, the giuen Arke A. F. is 50. degrées, and the complement F. B. is 40. degrées, both which being added together doe make vp the whole Quadrant of 90. degrées, marked with the letters A. F. B. Now Sinus complementi is the crosse line marked with the let∣ters F. K. the totall Sine which is the whole Semidiameter and greatest right Sine, is marked with the letters B. E. But because it is not enough to know the signification of the things aboue spe∣cified to vse the foresaid tables when néede is, vnlesse you know

also how to find out those things in the said tables, I thinke it good therefore to shew you the order of the said tables by describing the same as followeth.

You haue then to vnderstand that the tables of Monte Regio printed in Folio, are contained in 18. Pages, and euery Page containeth eleauen partitions, called collums, whereof the first on the left hand containeth 60. minutes, which are to be counted from head to foote, as they stand in order one right vnder another in se∣uerall places, procéeding from 1. to 60. The second collum con∣taineth Sines. The third containeth onely a portion or part of one second, and from thence forth procéeding towards the right hand all the other collums doe containe in like manner Sines and the portion of one second. And right ouer the head of euery Sine (the first collum of Sines onely excepted, hauing nothing but a Cypher ouer his head) are set downe the degrées of the whole Quadrant called arches, in such order as from the first Page to the last, there are in all 89. degrées, or arches, as by perusing the said tables you may plainely sée. Now to find out in these tables the things aboue mentioned, you must doe as followeth.

First to find out the right Sine of any giuen Arke, you must séeke out the number of the said Arke in the front of the tables, and if the giuen Arke hath no minutes ioyned thereunto, then the first number of Sines right vnder the said Arke, is the right Sine thereof. But if it hath any minutes ioyned thereunto, then you must séeke out in that Page, where you found the giuen Arke, the number of the minutes in the first collum of the said Page, on the left hand, and right against those minutes on the right hand, in the square Angle right vnder the said arch, you shall find the right Sine. As for example, you would find out the right Sine of a gi∣uen Arke containing 8. degrées, and 20′· here hauing found out in the front of the second Page the figure of 8. standing right ouer the eight collum, séeke in the first collum on the left hand of the said Page, for 20. minutes, and right against the 20. minutes you shall find on the right hand in the common Angle or square 8695 93. which is the right Sine of the foresaid giuen Arke, so as you make 6/000/000. to be the totall Sine: but if you make 60/000. the totall Sine, then you must alwaies reiect the two last figures standing on the right hand of the said right Sine, and the rest of the figures shall be the right Sine.

Now to find out the complement, there is nothing to be done, but onely to subtract the giuen Arke out of the whole Quadrant which is 90. degrées, and the remainder shall be the complement: as in the former example by subtracting 8. degrées, 20′· out of 90 degrées, you find that there remaineth 81. degrées, 40′· which is the complement of that arch. Againe to find out the Sine of the complement you must doe thus, séeke the complement in the front of the tables of Sines, euen as you doe to find out any giuen arke: as in the former example, the complement being 81. degrées 40′· you must séeke 81. in the front of the 17. Page of the first tables, which being found, séeke out also the 40′· in the first collum of the said Page on the left hand, and right against those 40′· in the com∣mon Angle right vnder the Arke 81. you shall find 5/936/649. which number is the right Sine of the foresaid complement, so as you make 6/000/000. to be the totall Sine, for if 60/000. be the total Sine, then you must reiect (as I said before) the two last fi∣gures on the right hand, and the number remaining shall be the right Sine of the foresaid complement, and therefore in working by these tables, you must alwaies remember what number you make the totall Sine to be.

Sinus versus commeth seldome in vse, notwithstanding if you would know how to find it out, you néede to doe no more but to sub∣tract Sinum complementi of the giuen Arke, out of the totall Sine, and the remainder shall be Sinus versus, as in the former example your Sinus complementi was 5/936/649. which being subtracted out of the totall Sine 6/000/000. there remaineth this 63/351. and that number is Sinus versus: for if you adde this remainder to the number which you subtracted, it will make vp the totall Sine 6/000/000. But there is one thing more necessa∣rie to be knowen then this, because it commeth oftner in vse, and that is vpon some diuision made how to find out the Arke of any quotient, which is to be done thus: Enter with the quotient into the body of the tables, and leaue not séeking amongst the squares of the Sines, vntill you haue found out the iust number of the quo∣tient (if it be there) if not, you must take the number of that Sine which is in value most nigh vnto it, whether it be a little more or lesse, it maketh no matter, and hauing found that number, looke in the front of that collum, and you shall find the Arke of your quoti∣ent,

standing right ouer the head of that collum, and also the my∣nutes thereof in the first collum of the said Page on the left hand. As for example, hauing diuided one number by another, I find the quotient to be 469/012. whereof I would know the arch, now in séeking this quotient amongst the Sines, I can not find that iust number, but I find in the first Page, and in the tenth collum 469/015. which is the nighest number vnto it that I can sée. In the front of which collum I find the Arke to be 4. degrées, and di∣rectly against that Sine on the left hand, I find 29′· belonging to that arch, whereof that quotient is the Sinus, so as I gather here∣of that the arch of the foresaid quotient is 4, degrées, 29′· But you haue to note by the way that the number of your quotient must ne∣uer be much lesse then 1745. for otherwise it is not to be found in these tables, vnlesse you make the totall Sine to be but 60/000. for then by reiecting the last two figures on the right hand, as I haue said before, the first right Sine of these tables shal be no more but 17. and by that account a very small quotient may be found in these tables. And whatsoeuer hath béene said here touching the or∣der that is to be obserued in the first tables of Monte Regio, whose totall Sine is 6/000/000. the like in all points is to be obserued in the last tables, whose totall Sine is 10/000/000. Thus much touching the order of the foresaid tables of Monte Regio Printed in Folio: but for so much as those tables bee not altogether truely Printed, and for that they haue béene lately corrected, and made more perfect by Clauius, who doth set downe the said tables in quarto and not in folio, whereby they are the more portable, and the more commodious, as well for that they are more truely Prin∣ted, as also for that the complement of euery Arke is set downe in euery Page at the foote of euery collum, so as you néed to spend no time in subtracting the Arke from 90. I thinke it good therefore to make a briefe description of those tables, and the rather for that I haue requested the Printer to print the like here in quarto, and I doe worke all such conclusions as hereafter follow, by the said tables, the totall Sine whereof is 10/000/000. according to the last tables of Monte Regio. But for so much as some may haue already the tables of Monte Regio Printed in Folio, not knowing perhaps the vse thereof, I will set downe two conclusi∣ons to bee wrought by those tables, and all the rest of the conclusi∣ons

are to be wrought by those tables which I haue here caused to be Printed in quarto like to those of Clauius: and though the two conclusions next following, which are to shew the vse of the fore∣said tables, may be wrought by the tables of Sines in what forme so euer they be truely Printed if Folio, or in quarto, yet because I had appointed them to bee done by the tables of Monte Regio, Printed in folio before that euer I saw Clauius his booke, I mind not now to alter them but to let them stand still as they are.