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 Astronomicall Fractions. Cap. 27

BEcause the vse of these Fractions is very necessarie for those that haue to calculate the motions of the Starres and the diffe∣rence of time, I thought good to shew here how the same are to be added, subtracted, multiplyed and diuided, for the measure of time falleth not out alwaies to bee a whole yeare, moneth, day, or hower, nor the mouing of the celestiall bodies are to be measured alwaies by whole circles, signes, or whole degrées, & therfore to haue an exact measure of such things it was thought best by the auncient writers to diuide all whole things called In∣tegra into the lessest parts that might bee, for which purpose no number was thought so méete as 60. for there is no number vn∣der

100. that receiueth so many Diuisions as 60. which may bee diuided many sundrie waies, that is by 2. 3. 4. 5. 6. 10. 12. 15. 20. and by 30. and therefore they diuided euery whole thing that had no vsuall parts into 60. minutes, and euery minute into 60. se∣conds, and euery second into 60. thirds, and so forth vnto 60. fourthes, fifts, sixts, seuenths, eights, ninthes and tenths, and further if néede were but that seldome chanceth. And you haue to note that minutes are marked with one stréeke ouer the head, se∣conds with two stréekes, thirds with thrée stréekes, and so forth thus, 23′· 6″· 7‴· 8''''· &c. which do signifie 23. minutes 6. seconds 7. thirds, and 8. fourths.

Of Addition.

VVHat is to bee obserued in adding of these kindes of Fractions?

First that you bring Integrums to Integrums, and Fracti∣ons to Fractions, that bee of like denomination, beginning al∣waies with the least on the right hand, and if the summe of such Addition doe amount any where to the number of 60. or aboue 60 then you must looke how many 60. are comprehended therein, and for euery 60. adde one to the next greater Fraction that is on the left hand, obseruing still that order vntill you come to the In∣tegrums, of which Integrums, it is also necessary to know their values, that is to say, what parts they containe, and what denomi∣nation those parts haue: As for example, if you adde common signes such as the twelue signes of the Zodiaque be, then euery signe containeth 30. degrées, so as euery summe excéeding 30 is to be diuided by 30. but if they be Phisicall signes whereof 6. doe make a whole circle, such as be set downe in the table of Alphon∣sus, then the summe of those degrées is to be diuided by 6. More∣ouer so often as the summe of the common signes doe excéede 12. or the summe of the Phisicall signes doe excéede 6. the ouerplus is alwaies to be reiected, and the remainder to be set in the place of the signes, as you may sée in this example following wherein se∣conds are reduced to minutes, minutes to degrées, and finally de∣grées, to signes.

An example of Addition consisting of signes, degrees, minutes and seconds.

	Signes	Degrees	Minutes	Seconds
9.	19.	1.	19.
0.	0.	35.	16.
11.	29.	33.	5.
9.	29.	38.	11.
0.	11.	49.	40.
		4.	56.
The totall summe.	8.	0.	42.	27.

In which example beginning first with the seconds because they are here the least Fractions, you shall finde by Addition that the summe of them a∣mounteth to 147. which being diuided by 60. you shall finde in the quotient 2. and the remainder to bee 27″· which remainder you must set downe vnder the collum of seconds, kéeping the quo∣tient which is 2. in mind to be added to the collum of minutes, the summe whereof is 162. which being diuided by 60. you shall find in the quotient 2. degrées, and the remainder to be 42′· which is to be set downe vnder the collum of minutes, and the quotient 2. kept in minde to bee added to the collum of degrées, the summe whereof is 90. degrées, which being diuided by 30. (because that 30. degrées doe make one common whole signe) you shall find in the quotient 3. signes and no remainder: wherefore you must set downe a Cypher vnder the collum of degrées, and adde the 3. signes to the collum of signes, the summe whereof is 32. which it you diuide by twice 12. which maketh 24. you shall finde the re∣mainder to be 8. signes, which is to be set downe vnder the collum of signes, for the quotient here is to be reiected according to the rule before giuen, so as the totall summe of this Addition is 8. signes, 0. degrées 42′· 27″·

Another example of daies, howers, minutes and seconds to be added together.

	Daies	Howres	Minutes	Seconds
21.	14.	32.	11.
16.	16.	19.	41.
8.	16.	30.	30.
Summa to∣talis.	46.	23.	22.	22.

In this example one self order as in the other before is to bée obserued as touching the secondes and my∣nutes, for the excée∣ding

number in both, are diuided by 60. but when you come to the howers you must diuide that number by 24. because that so many howers doe make a whole day, and hauing set downe the re∣mainder vnder the collum of howers, adde that one day which was in the quotient, vnto the collum of daies, and so you shall find the totall summe to be as the example aboue sheweth.

Of Subtraction.

VVHat order is to be obserued in Subtraction?

The selfe same that was before obserued in Addition, so as you alwaies remember that when you haue to take a greater number of Fractions, as of minutes, seconds, thirds and such like, out of a lesser number of Fractions, to borrow 60. and ha∣uing set downe the remainder to adde the one borrowed, vnto the next collum on the left hand, for there the 60. borrowed, is but one, but if you haue to deale with degrées, which are counted In∣tegrums, then you must borrow but 30. for so many degrées doe make one signe, and if you haue to subtract howers, then you must borrow 24. for so many howers doe make one day, as by the example here following you shall more plainely perceiue.

Signes	Degrees	Minutes	Secondes
8.	0.	42.	27.
0.	1.	9.	53.
7.	29.	32.	34.

In this example because you can not take 53″· out of 27″· you must borrowe one minute from the next collum on the left hand, which one my∣nute is 60″· which being added to 27. doe make in all 87″· out of which if you take 53″· there will remaine 34″· that done, you must pay home the minute which you borrowed by adding the same vn∣to the next 9. on the left hand, which maketh 10. then say, take 10′· out of 42′· and there remaineth 32′· which being set down, pro∣céede to the next. But here to take one out of none that cannot be, and therefore borrow one whole signe from the collum of signes which is 30. degrées, from whence take one, and there remaineth 29. which being set downe, take the one which you borrowed out of 8. and there remaineth 7. so as the whole remainder of this Sub∣traction

is 7/ signes, 29/degrées, 32′ / and 34″· as the former ex∣ample sheweth.

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.

Of the Diuision of Fractions Astronomicall.

VVHat is to be obserued therein?

First you must consider whether your Diuisor be com∣pound, or simple, I cal that compound which containeth Fracti∣ons of diuers denominations, and that simple which consisteth of Integrums, or is one whole number of one selfe denomination, wherein there is no dificultie, for then you haue no more to do but to diuide euery particular number contained in the diuidend by y^e same Diuisor and to place the product of euery one vnder such de∣nomination, as the little table of denominations sheweth, & there∣fore it shall not bee amisse to set the foresaid little table ouer your diuidend euen as you did in Multiplycation: Also the Sexagenary progression is alwaies to be vsed, as well in Diuision as in Multi∣plycation. Moreouer if your Diuisor be not exactly contained in y^e diuidend, then hauing multiplyed the diuidend by 60. you must adde to the product therof the next Fraction following: As for ex∣ample, knowing by Alphonsus tables that the daily motion of the Moone is 13. degrées, 10′· 35″· 1‴· 15''''· you would know how much the goeth in the space of an hower, here because that one day con∣taineth

24. howers, the number must be 24. your Diuisor which is simple and not compound, first then set downe in the front of your worke the rowe of denominations onely, and not the natural numbers, because they are not to be vsed in this way of Diuision, that done, right vnder the rowe of denominations place your di∣uidend, and right vnder y^• your Diuisor, as you sée in this example.

Degree••	′	″	‴	''''	v	The denominations.
13	10	35	1	15		The diuidend.
24						The diuisor.
	32	56	27	33	7	The seuerall summ of euery quotient.

In which example because the Diuisor 24. is not contained in 13. therfore I mul∣tiply 13. by 60. which maketh 780. where∣vnto by adding the next Fraction on the right hand which is 10′· the whole summe is 790′· which being diuided by 24. the quoti∣ent is 32′· which because they are minutes, I place them vnder the denomination of minutes, and the remainder is 22′· which being multiplyed by 60. maketh 1320″· wherunto I adde the next figure which is 35. and so the whole summe is 1355″· which being diuided by 24. the quotient is 56″· which I place vnder the denomination of seconds, and the remainder of this Diuision is 11″· which being multiplyed by 60. maketh 660‴· whereto I adde the next Fraction which is 1‴· so that now the whole summe is 661‴· which being di∣uided by 24. the quotient is 27‴· which I set downe vnder the de∣nomination of thirds, and the remainder is 13‴· which being mul∣tiplyed by 60. maketh 780''''· whereunto I adde the next Fraction which is 15''''· which maketh in all 795''''· which being diuided by 24. the quotient is 33''''· which I place vnder the denominatiō of fourths. and the remainder is 3''''· which being multiplyed by 60. maketh 180^v· whereunto hauing no Fraction to adde, I diuide the same by 24. and so I find in the quotient 7^v· which I set vnder the denomi∣nation of fifts, so as I find the howerly motion of the Moone to bée 32′· 56″· 27‴· 33''''· 7^v· and somewhat more, for I leaue to deale any fur∣ther with the smaller Fractions that would stil grow by multiply∣ing the remainders by 60. thinking this sufficient to shew you in what order you haue to worke, to diuide your diuidend by a simple Diuisor, into as many small parts as you will: but if your Diui∣sor be compound, then the Diuision is to be done either by reducti∣on into the smallest Fractions, or without reductiō: which last way

is very hard and tedious, and therefore I will onely shew you how to make your diuision whereof the Diuisor is compound by reduc∣tion, and that by this one example here following. Suppose then that the Moone according to her owne course which is from West to East, is distant from some fixed Starre 36. degrées. 30′· 24″· 50‴· and 15''''· and that you would know in what time she will runne that distaunce, according to her daily moouing which as hath béene said before is 13. degrées, 10′· 35″· 1‴· and 15''''· here to make this di∣uision by reduction, you must doe thus. First reduce all the num∣bers of your diuidend into the smallest Fractions thereof by the Sexagenarie Multiplycation and Addition of the next Fraction vnto the product of that Multiplycation: that done, reduce all the numbers of your Diuisor by like Multiplycation and Addition, into the smallest Fractions, so as the diuidend & the Diuisor may be both of one selfe denomination, and diuide the one by the other, euen as they were Integrums, as in this example you must first multiply 36. degrées, by 60. and it will make 2160′· whereto by adding 30′· you make the whole summe of minutes to be 2190′· which being multiplyed againe by 60. doe make 131400″· whereto if you adde the 24″· the summe of secondes will bee 131424″· and so procéeding still with the Sexagenarie Multiplycation and Addi∣tion of the next Fraction as you did before, you shall find the diui∣dend to be 473129415''''· Then in like order reduce your Diuisor in∣to the smallest Fraction, and you shall find the totall summe ther∣of to be 170766075''''· this reduction being made, diuide the diui∣dend by the diuisor, so shal you find in the quotient 2. Integrums, that is to say 2. daies, and the remainder to be 132597365''''· which remainder if you multiply by 60. and diuide the product by the self same Diuisor, you shal haue in the quotient minutes, then mul∣tiply againe that remainder by 60. and diuide the product thereof by the same Diuisor, and you shall haue in the quotient seconds, and so by obseruing still that order you shall bring it into as smal Fractions as you will, thus shall you finde that the Moone accor∣ding to her daily motion, will runne the foresaide space of distance that was betwixt her and the fixed Starre in 2. daies, 46′· and 14″·

Notes

• The re∣mainder.