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.

Of Multiplycation.

THough there bee more difficultie in multiplying and diuiding Astronomicall Fractions then in adding or subtracting them, yet the greatest difficultie thereof chiefly consisteth in the finding out of the true denomination of the products, for first as touching Multiplycation you must multiply euery number of the multiply∣er into all the particular nūbers of the sum that is to be multiply∣ed, and then seuerally to adde together the products that bee of one selfe denomination, and whatsoeuer in that Addition ariseth to the number of 60. or excéedeth 60. it is to be reduced by the sex∣aginarie Diuision into the greater summe, so shall you collect the whole summe of the Multiplycation. But you haue to note by the way, that if there be any Integrums of diuers denominations in your multiplyer: That then such Integrums must be reduced to one selfe kind of Integrums: As for example, suppose that you would multiply the daily motion of the Moone which according to Alphonsus tables, is 13. degrées, 10′· 35″· and 1‴· by 29. daies, 12. howers, 44′, and 3″· here because there be in this multiplyer Integrums of diuers denominations, that is to say daies and howers, you must therefore reduce the same into one selfe denomi∣nation, before that you can make your Multiplycation.

How is that to be done?

By multiplying euery number of the said multiplyer by 5. and then by halfing the product thereof, by which halfing you shall re∣duce mynutes to seconds, and seconds to thirds, and so forth to the smallest Fractions of all, and if any product doe amount to 60. or excéede 60. then you must diuide that product by 60. the ••emainder whereof is to be set vnder his proper denomination, and you must kéepe the quotient in minde to adde the same to the next greater number as this table sheweth: In the front whereof I haue set downe in seuerall spaces not onely the denominations of the two Integrums, as daies and howers, but also the denomi∣nations of so many Fractions as I think méete to serue my turne, vnder which from I place the foresaid multiplyer, and then draw a line as you sée in this example following.

Denomination.	Daies	Howers	′	″	‴	''''
The multiplyer to be reduced into one selfe denomination.	29.	12.	44	3
The products of the reductiō.		29.	31	50	7	30

Now beginning first on the right hand with the least Fraction of the saide multiplyer which is 3″· I multiply 3. by 5. which ma∣keth 15. the one halfe whereof is 7‴· and halfe a third which is 3''''0· wherefore I set downe the said 7‴· and 3''''0· vnder their proper de∣nominations as you sée in the example aboue, then I multiply 44′ by 5. the product whereof is 220. and the halfe thereof is 110. which being diuided by 60. the quotient is 1′· and the remainder 50″· which remainder I set vnder his proper denomination, kée∣ping the quotient stil in mind, that done, I multiply 12. howers by 5. the product whereof is 60. and the halfe of that is 30′· where∣vnto by adding the one which I had in mind, I make it 31′· and so I set it downe vnder his proper denomination, and because there be no more Fractions to be multiplyed, I set downe on the left hand the Integrum 29. and by this meanes I haue brought the foresaid multiplyer to one selfe denomination and to one kind of Integrum, that is to say to 29. daies 31′· 50″· 7‴· 3''''0· which now being the greater number is to be set aboue, and to be Multiplyed by the foresaid daily motion of the Moone, that is 13. degrées, 10′ 35″· and 1‴· but to the intent that in multiplying these numbers to∣gether you may set euery product in his true place, that is to say, vnder his proper denomination, it shall not be amisse in the front of your worke to set downe two rowes of numbers, whereof the first must containe many denominations or Fractions as you thinke good, as minutes, seconds, thirds, fourths, and so forth, marked with stréekes and vulgare numbers, and the second rowe shall be the naturall order of numbers written in Arithmeticall fi∣gures as this table sheweth.

Integra	′	″	‴	''''	v·	vi·	vii	viii	The denominations.
0.	1.	2.	3.	4.	5.	6.	7.	8.	The naturall numbers.

Vnder which table you must first set the number that is to bée multiplyed, and right vnder that the multiplyer in such sorte as

euery particular product may be placed vnder his proper deno∣mination, and then draw a line as you sée in this example follow∣ing, and when you haue multiplyed 2. numbers the one into the other, and know not where to place the product, then marke vn∣der which of the naturall numbers in the front, the said two num∣bers, that is to say the multiplyer and the multiplycand do stand: That done, adde those two naturall numbers of the front toge∣ther, and the summe thereof will them you vnder what denomina∣tion the product is to be placed, as in this example.

Int••gra	′	″	‴	''''	v	vi	vii	Denomination
0	1	2	3	4	5	6	7	Naturall num∣bers.
29	31	50	7	30				The multiply∣cand.
13	10	35	1					The multiplyer
			29	31	50	7	30	The seueral pro∣ducts.
		1015	1085	1750	245	1050
	290	310	500	70	300
377	403	650	91	390
389	6	24	2	31	12	37	30	The general ••roduct or totall summe.

In which example I first multiply 1‴· into 3''''0· the product whereof is 3^vii0· which must be set vnder the denomination vii. be∣cause the two naturall numbers that is 3. standing ouer 1. the multiplyer, and 4. standing ouer 30. the multiplycand being ad∣ded together doe make 7. appointing to the product his proper denomination, then multiply againe the same 1‴· into 7‴· the pro∣duct whereof is 7^vi· which must be set vnder the denomination vi. because 3. standeth ouer both their heads, and therefore must bée taken twice, that is to say for each number 3. which being added together doe make 6. appointing thereby to the product his pro∣per place of denomination, that done, multiply the said 1‴· into 50″· the product whereof is 50″· which must be set vnder the denomina∣tion v. for the 3 which standeth ouer 1. and the 2. ouer 50″· being added together maketh 5. appointing to his proper place of deno∣mination, then multiply the said 1‴· into 31′· the product whereof is 1''''1· here the two naturall numbers that is to say 3. standing ouer 1‴· and one ouer 31′· being added together, doe make 4. ap∣pointing to the product his proper place of denomination, then multiply the foresaid 1‴· into 29. Integrums, the product whereof

is 2‴9· and must be set vnder the denomination ‴ because 3. standeth ouer 1‴· but 29. being an Integrum, hath no natural number stan∣ding ouer him but a Cypher, thus hauing gon through out all the numbers of the multiplycand, with the first number of the multi∣plyer, procéede in like order with the second number of the multi∣plyer, which is 35″· which being multiplyed into 3''''0· maketh 10^vi50· to be set vnder the denomination vi. because 2. standeth ouer 35″· the multiplyer, and 4. ouer 3''''0· the multiplycand, which 4. and 2. being added together maketh 6. then multiply 35″· by 7‴· which ma∣keth 245^v· which you must set downe vnder the denomination v. be∣cause 2. and 3. maketh 5. that done, multiply 35″· by 50″· which maketh 1750''''· which you must set downe vnder the denomination '''' because the multiplycand and the multiplyer are both vnder the denomination ″ which being twice repeated, maketh 4. then multiply 35″· by 31′· and that maketh 1085‴· which you must set downe vnder the denomination ‴ because 1. and 2. maketh 3. fi∣nally multiply 29. Integrums, by 35″· and that maketh 1015″· thus as you haue gon through with two numbers of your multiplyer, so procéede in like order with the other two numbers of the multi∣plyer which is 10. and 13. and when you haue ended your Mul∣tiplycation, and set euery product in his proper place, and so as euery figure may stand one right vnder another, to auoide confu∣sion when you come to Addition, (to which end the spaces of col∣lums had néede to be the larger) then draw a line vnder all the pro∣ducts and beginning on the right hand, adde all the products con∣tained in euery seuer all collum together, and if the summe of any such particular Addition do arise to the summe of 60. or excéed the number of 60. then diuide that summe by 60. and set downe the remainder, kéeping the quotiēt in mind to be added to the product of the next collum on the lefthand, so shall you find the total summe of your Multiplycation to be 389. degrées, 6′· 24″· 2‴· 31''''· 12^v· 37^vi· and 30^vii· as the former example plainly sheweth. Now if you dsuide 389. degrées, by 30. because euery common signe containeth 30. degrées, you shall find your totall summe to be 12. signes, 29. de∣grées, 6′· 24″· 2‴· 31''''· 12^v· 37^vi· and 30^vii· and so much the Moone runneth in the space of 2. 9. daies 12. howers, 44′ and 3″· of an hower, which is her full reuolution betwixt euery two changes, but for as much as it chanceth as wel in this example as in many others like, that

Integrums of two sundrie denominations are propounded in the question, it may be very well doubted with what denomination the product of such multiplycation is to be named, as in this example hauing multiplyed time by motion, a man may aske whether the product shall be named daies or degrées, the resoluing of which doubt dependeth vpon the nature of the question propounded, for in the foresaid example, because time or daies do comprehend any certaine appointed motion, therefore the product of the Multiply∣cation is to be referred to the degrées of motion which are compre∣hended vnder time, and not to time which comprehendeth motion, wherfore this product of Integrums videliz. 389. signifieth here degrées and not daies, so likewise when degrées and minutes are multiplyed by myles and minutes, the product of such Multiply∣cation taketh his name frō myles & not from degrées, because de∣grées do comprehend myles, for we say in matters of Geography that euery degrée of the great circle comprehēdeth 60 myles, thus hauing spoken sufficiently of the Multiplycation of Astronomical Fractions, we wil now procéede to y^e Diuision of such Fractions.

Notes

• The re∣mainder.