Selected quad for the lemma: day_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
day_n	london_n	mile_n	minute_n	5,572	5	12.4321	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A96647	Willsfords arithmetick, naturall, and artificiall: or, decimalls. Containing the science of numbers, digested in three books. Made compendious and facile for all ingenious capacities, viz: merchants, citizens, sea-men, accomptants, &c. Together with the theorie and practice united in a sympathetical proportion betwixt lines and numbers, in their quantitites and qualities, as in respect of form, figure, magnitude and affection: demonstrated by geometrie, illustrated by calculations, and confirmed with variety of examples in every species. / By Thomas Willsford, Gent.	Willsford, Thomas.; Vaughan, Robert, engraver.	1656 (1656)	Wing W2874; Thomason E1584_1; ESTC R204128	162,697	364

but if the Progression begins with any other figure that is the Progressions terme then multiply that terme by the number of Progressions the product will be the extreme required and the contrary if the greater extreme be given divide it by the number of Progressions the Quotient will give the lesser extreme as if 2 were the lesser extreme and the Progression what will the 6 place be The answer to this is 12 thus they will stand 2 4 6 8 10 12 if by 3 what the 7 place the answer will be 21 as it will thus appear 3 6 9 12 15 18 21 if the greater extreme were given to finde the lesse as 32 whose Progression is to 8 places therefore divide 32 by 8 the Quotient is 4 the lesse extreme and the order of Progression as will thus appear 4 8 12 16 20 24 28 32 or 30 the greater extreme in the sixt place the lesser extreme was 5 but when the first terme is not the difference of the Progression then multiply the term of the Progression given by one place lesse than the number of Progressions and to the product adde the lesse extreme the totall will be the greater extreme as if 3 were the lesse extreme of a Progression by 2 unto 7 places then say 6 times 2 is 12 to which adde 3 the lesse extreme the summe will be 15 for the greater and thus appears 3 5 7 9 11 13 15. The greater extreme with the number of Progressions and their difference being given to finde the lesser extreme observe this as a generall Rule multiply the difference of Progressions by one lesse than the number of the said Progressions subtract the product from the greater extreme given and the remainder will be the first number of that Progression continued As for example admit 44 were the extreme propounded which had proceeded by 10 in five Progressions 10 multiplied by 4 produceth 40 which taken from 44 there will remain 4 for the first number and the lesser extreme as by these numbers will appear viz 4 14 24 34 44. Again admit 29 to be the greater extreme proceeding by 2 in 13 Progressions I say 12 multiplied by 2 produceth 24 which subtracted from 29 leaveth 5 for the first number as 5 7 9 11 13 15 17 19 21 23 25 27 29 and so the like of any other continued Progression With a mean proportionall and one extreme being given to finde the other extreme and the number of progressions Example 4. Any numbers in Arithmeticall proportion the summe of the extremes is double unto the mean by the former Theoremes and therefore it is evident that either of the extremes taken from the meane doubled will give the other extreme as 4 6 8 and 6 doubled is 12 from whence take 4 and there will remain 8 or subtract 8 and there will remain 4 and consequently the two extremes being given the meane is also included being halfe the summe of two extremes and to finde the number of Progressions subtract the lesser from the greater and divide the remainder by the difference of Progressions the Quotient will be alwayes one lesse than the number as if 5 29 were two extremes whose difference was 2 the difference between them is 24 which divided by 2 the difference of Progression the Quotient will be 12 unto which adde one the number is 13 the whole series or order of the Progression viz 5 7 9 11 13 15 17 19 21 23 25 27 29. For the clearer explanation of this suppose two men were to goe from London to Edenborough by computation 292 miles they took horse together one resolving for to travell 30 miles every day the other 6 miles the first day the second day 12 and so proceeding in Arithmeticall Progression of 6 untill he should arrive at his journeys end and here it is required to know which of them came first to Edenborough or in how many dayes one will overtake the other first observe that 30 is here the mean proportionall and 6 the lesser extreme and the greater 54 the mean doubled is equall unto the summe of the extremes that is 60 as by the former Rules where the number of progressions will be found 9 and in so many dayes these Travellers will overtake one the other having gone 270 miles which is 9 times 30 equall to the other as by these numbers 6 12 18 24 30 36 42 48 54 in this series you may observe 30 the mean exceeding the others first dayes journey as it is exceeded by the last and consequently first at his journeys end for the next day will be 60 miles wherein he rides as fast again as the other and having but 22 miles to ride I will leave them and end this Paragraph and proceed no farther in Naturall Progressions Paragraph VI. Treating of Geometricall Progressions and Proportions continued and interrupted with the addition of those numbers GEometricall Progression doth consist of more than two numbers not of equall difference in numbers but of like proportion in quality and quantity viz 1 2 4 8 16 32 64 c. this is called a double proportion every one being but half the succeeding number or these 1 3 9 27 or 2 6 18 54 denominated a triple proportion every number containing the precedent 3 times so 2 8 32 128 or 3 12 48 a quadruple proportion every succeeding number being 4 times the precedent and so of all other numbers of what quality or quantity soever proceeding in this kinde and is nominated Geometricall proportion continued Proportion interrupted or broken off is when the Progression is discontinued in the proportion of the numbers and consists of four places or progressions at the least viz 3 6 7 14 wherein the second and the third number that is 6 7 differs in the proportion not as 3 to 6 or 7 to 14 which is a double proportion and so 1 3 27 81 is a Geometricall progression interrupted for 1 3 27 81 are in a Triple proportion but not the second and the third 27 containing 3 nine times Theoreme 1. Geomet●icall progressions are either ascending or descending in the former proportions and if continued and proceeding from an unite ●he first from 1 is called the Root or first quantity The second is the Square or the second quantity The third the Cube or third quantity The fourth is Biquadrat the Squared square or fourth quantity The fift is called the Surde solid or fift quantity The sixt is denominated the Squared Cube or the sixt quantity c. all which are explained in these numbers viz 1 being neither Number nor Quant●ty 2 the Root 4 the Square 8 the Cube 16 the Biquadrat made by the square of 4 next 32 is the Surde solid made by the multiplication of the Biquadrat and the Root or the product of the Square and the Cube 64 is the squared Cube that is the Square made of the Cube multiplied in it selfe all which

in their own figures and from thence are called Circular numbers 7 multiplied by 7 ends in 9 3 1 and fourthly in its self 8 by 8 ends in 4 2 6 and fourthly in its selfe again 9 multiplied by 9 will end in 1 and in every second multiplication is terminated in 9 it s own root or number again produced Multiplication is a quadrature and hath this Analogy or proportion with a superficiall square call'd in Geometrie the second quantity a figure composed of lines whose sides are divided into parts and intersected with paralle's or equidistant lines as is the last figure A B C D making thus both Squares and right angled Parallelograms equall to the numbers multiplied in themselves together as for example the Table or square A C hath every side divided into 9 equall parts as admit Poles Yards Feet Inches or what you please suppose these sides as A B and A D in Feet and intersected with lines the whole Square will containe 81 square Feet for the true superficiall content of it so if 9 were multiplied by 9 the Product will be the same or if the long Square A F E D were given the superficiall content of this Geometricall figure call'd a Parallelogram would be 45 square Feet and so is the product of 9 multiplied by 5 If it were required to know how many feet there were in a Yard square 3 Feet makes a yard in length therefore if every side of such a square were divided into 3 equall parts and intersected with right lines there will be found 9 square feet as in the Table will appeare and so 3 multiplied by 3 will produce 9 and in these the proportion is continued as an unite is unto the number or side of the Square given so will the side be to the whole Square or as 1 to the Multiplier so will the Multiplicand be unto the Product as in the former examples as 1 to A B 9 so will A D 9 be to 81 the whole Square or Product or as 1 to A F 5 so A D. 9 in proportion to 45 the long Square A F E D and so the like of any other in this kind whether greater or lesse and so much for the Sympathy betwixt Multiplication and Geometricall Squares of the second quantity The way and form of Multiplication when there is more than one significant figure in the Multiplicand Multiplier or in both of them When there be severall figures to be multiplied both in the Multiplier and Multiplicand set down first the greater number and under that the lesser according to their degrees or places as unites under unites tennes under tennes hundreds underneath hundreds c. as hath been directed in divers other examples already this done draw a line underneath them both whereby to separate them from the numbers encreasing by their mutuall multiplications proceeding from the right hand in order towards the left every figure of the Multiplier must be encreased or multiplied into every particular figure of the Multiplicand from whence there will arise so many Products as there be figures in the Multiplier the first figure in every Product ought to be placed exactly under that figure which multiplies and so in order with a convenient distance towards the left hand for all the severall Products must be added together whereby to finde the result or totall of them and in every figure as you multiply set down the Product if under 10 but if a decimall or decimalls subscribe a cypher if a compound number write down the significant figure and keep the decimall or decimalls in your minde and as you multiply the next figure adde them in as unites and so proceed untill every figure of the Multiplier be encreased by all the figures of the Multiplicand in order as was said already and shall be now illustrated with severall Paradigma's following To finde how many dayes there be contained in a common Year consisting of 52 Weeks and one day Example 1. Having set down the 52 Multiplicand 52 The Multiplier 7 The Product 364 The odde day 1 The totall of days in a year 365 Weeks subscribe underneath it the dayes in one Week which are 7 and being a simple number I set it under 2 in the unite place and since there are so many dayes in one week there must be 52 times so many in a yeare besides the odde day then say 7 times 2 makes 14 for which subscribe 4 below the line and right under 7 and keep the decimall in your minde they say 7 times 5 as by the Table of Multiplication will make 35 and the 1 decimall in minde makes 36 therefore subscribe 6 right underneath the 5 and keep the 3 decimalls in minde but since there is no more write it down and the product is 364 the number of dayes in 52 weeks and being there is one day more in a yeare adde that in and then the totall will be 365 the true number of dayes in the Suns annuall revolution the thing required The dayes and parts of a yeare being known to finde the number of houres contained in it Example 2. First set down the greater Multiplicand 365 Multiplier 24 The Products 1460   730 The totall 8760 number and then the Multiplier according to the places of the figures in this 24 being the houres contained in a naturall day and then say 4 times 5 is 20 for which subscribe a cypher and keep 2 decimalls in minde then 4 times 6 is 24 2 is 26 that is 6 and goe 2 then say 4 times 3 is 12 and 2 in minde is 14 and being there are no more figures in the Multiplicand write them down and then the first Product will be 1460 then goe to the second figure of the Multiplier which here is 2 and say 2 times 5 is 10 subscribe a cypher under 2 the Multiplier and goe 1 decimall then 2 times 6 is 12 and 1 makes 13 that is 3 and goe 1 then 2 times 3 is 6 and 1 in minde makes 7 which write down under the 1 and then adde the Products together the totall will be 8760 the number of hours in 365 dayes and since the Julian account makes the magnitude of the yeare for to consist of 6 hours more adde 6 unto the Product and the totall will be 8766 hours contained in a common year To finde how many minutes are contain'd in a year Example 3. Here you are to know the Multiplicand 8766 Multiplier 60   0000   52596 Product 525960 parts or minutes of an hour which are 60 those must be the Multiplier and 8766 the houres contained in a yeare the multiplicand then say no times 6 or 6 times nothing is nothing therefore subscribe so many cyphers as there be figures in the Multiplicand and then begin with 6 and say 6 times 6 is 36 there set downe 6 beneath the line right underneath the Multiplier 6 then say 6 times 6 makes 36 and 3 decimalls in minde will

28 the Divisor make a point under the second 6 from 1 and then see how many times 2 may be contained in 16 6 will be too great for 6 multiplied by 28 the product will be 168 whereas the 3 first figures are but 166 therefore take 5 for the first Quotient which in order multiplied by the former rules into the Divisor and the product subtracted from the Dividend the remainder is 26 before which make an other point under 3 the last figure of the Dividend and finde a new Quotient which in this will be 9 that multiplied and the product subtracted the true remainder will be 11 the number desired for the yeare 1654 by which Cycle the Dominicall Letter will be found A the Quotient 59 the number of those Circles revolutions since our Saviours Birth but here I will say no more of this lest I have said too much already intending here Arithmetick and not the computation of Kalenders but these two questions or propositions I rather chose to shew how ready for use the remainder stands being in these the thing chiefly required whereas in other questions excepting such as are of severall denominations the remainder will be unknowne without the knowledge of Fractions A number of divers denominations given for to be divided into any parts required Example 9. Dayes Hours ′ ″ ‴ ' ' ' '   730 11 37 0 0 0 A The magnitude of two Yeares A Solar Moneth contains 30 10 29 2 30   B Here is given the space of 2 yeares and it is required for to know the greatnesse of a Solar moneth vulgarly accounted the space of 30 dayes a naturall day containing 24 houres every houre 60 minutes commonly noted with such a dash over it as thus 60 of them making one Second noted with 2 dashes of a pen every Second is divided into 60 parts call'd Thirds sign'd with 3 dashes Fourths with 4 c. There are in 2 yeares space 24 Moneths therefore the number given divided into 24 equall parts solves the question here propounded this may be effected by the fift Example but better by the way following in division of each particular part or denomination thus The number of dayes in 2 yeares are 730 which must be divided by 24 the number of moneths contained in that space of time having placed your numbers in order prescribed make a point under 3 in the Dividend which done you will finde 3 for the Quotient that multiplied into the Divisor and the Product subtracted from the Dividend the remainder will be found then make an other point a place farther as under the cypher and being 24 is not to be had in 10 put a cypher in the Quotient which is 30 dayes and the remainder 10 set down 30 in the former Table under dayes The last remainder was 10 dayes which convert into hours the next denomination lesse by multiplying of it into 24 the houres contained in a naturall day the Product will be 240 to which adde 11 houres being parts of the 2 yeares given the summe will be 251 hours which divide by 24 the Quotient will be 10 houres and the remainder 11 place 10 in the first Table against B under the title of Hours The last remainder was 11 houres which multiplied by 60 the minutes in one houre the Product is 660 to which adde 37 minutes against A the summe will be 697 for the Dividend which divided by 24 the Quotient will be 29 minutes and the remainder 1 set 29′ against B in the columne of minutes In the last operation there was but 1 minute remaining which is 60 seconds that divided by 24 the Quotient will be 2 seconds and the remainder 12 place the 2 in the first Table against B under the title of Seconds The 12 Seconds which did last remaine multiplied into Thirds the next denomination lesse the Product will be 720 which divided by 24 the Quotient will be 30 Thirds and nothing remaining if any thing bad you might have continued them in this manner to Fourths c. The last Quotient put into the Table you will finde against B that a Solar Moneth containes 30 Dayes 10 Hours 29 Minutes 2 Seconds and 30 Thirds Breviates or compendious wayes and observations in Division exemplified Example 10. If you are to divide by   5754 7101   1     4   2877 789     13701 10,000   2     3   4567 2500   any significant figure onely it is unnecessary for to set down the Divisor but to keep it in minde and the product as before while subtraction's made the Multiplier will be the Quotient in all such cases as by these 4 Paradigma's will appear the first is the yeare since the Creation of the World for to be divided into two equall parts make a point under 5 wherein 2 is twice contained and 1 remains then point forward 2 will be in 17 8 times then 2 in 15 7 times and lastly 2 in 14 7 times the Quotient 2877. In the second Table there is given 13701 Nobles one being in value 6 s. 8 d. 3 of them makes 20 s. and it is required to know how many pounds sterling it will make in all divide by 3 and say 3 will be found in 13 4 times in 17 5 times in 20 6 times and in 21 7 times the Quotient is 4567 li. In the third Example 10,000 Crownes are received beyond sea and it is required to know how many pounds sterling must be paid for them in England 4 of them making 20 shillings make a point under the cypher where 4 will be had in 10 2 times in 20 5 times and being nothing remains but 2 cyphers more towards the right hand annex them to it and it is done the Quotient being 2500 li. Fourthly there is 7101 li. for to be equally distributed unto 9 men 9 cannot be had in 7 therefore make a point under the next place and then 9 will be found in 71 7 times and next in 80 8 times and lastly in 81 9 times the Quotient 789 li. And so much does every one of their shares come unto When there is an unite with cyphers annext unto a Divisor cut off so many places upon the right hand of the Dividend as there were cyphers in the Divisor as for example 120 li. is to be distributed equally to 10 men for the cypher in the Divisor cut off one place in the Dividend and 12 li. will be every ones part or the Quotient or if 100 Souldiers were to receive for their pay 625 li. every ones share will be 6 li. and 25 li. over which by the former rules converted into shillings by being multiplied by 20 the product will be 500 s. from whence cut off the 2 cyphers and there will remaine 5 s. So 6 li. 5 s. every Soldier must receive And so for any other of this kinde the reason is evident for 1 divides nothing and the Quotient must have so many places

lesse than the Dividend by the number of cyphers in the right hand of the Divisor as by the common way of Division will plainly appear If the Divisor consists of any significant figure or figures in the formost place and a cypher or cyphers to the right hand leave out the cyphers in the Divisor and cut off so many places upon the right hand of the Dividend and with the residue divide and when the division is done annex the cyphers to the D visor and to the remainder the figures that were severed from the Dividend both of them constituting a Fraction or true part of the Integer Example 11. Admit the distance from London to the City of Yorke were 48,080 Poles or Pearches and it is required to know how many miles they are a sunder 40 Poles makes a Furlong and 8 of them a Mile or 320 Poles which is here the Divisor and cutting off the cypher on the right hand I must doe so in the Dividend which will be then 4808 for to be divided by 32 which accordingly done the Quotient will be 150 Miles and 8 remaining to which annex the cypher cut off it will be 80 Poles or a quarter of a Mile for the distance desired An Examine of Multiplication and Division These two species doe trie one the other as Addition and Subtraction did for in any number that is multiplied if the Product be divided by the Multiplier the Quotient will be the Multiplicand as before and so likewise any number that is divided if you multiply the Quotient and Divisor together and to the Product adde in the Remainder if there be any the summe or Product will be the Dividend again if your work be true Example 12. The Julian or old account An. Dō Bissextiles 1654 413   1652 413 2 Rem 2 1654 1 2 did make the magnitude of the year for to consist of 365 dayes and 6 houres 24 making a natural day for which cause every fourth year contained 366 dayes commonly called Leap-yeare or Bissextile in one of those was our sacred Redeemer borne now to finde this Bissextile for any yeare since or to come according to the old Kalender divide the yeare given by 4 the Quotient shews the revolution of those Leap-yeares since His happy Birth the remainder are the yeares elapsed since the last and if nothing remains it is Leap-yeare in this example is given the yeare 1654 which divided according to the tenth Example the Quotient is 413 the number of Leap-yeares past since the blessed Virgin Mary was a Mother and the Remainder is 2 and so many yeares are elapsed since the last Bissextile as by the first example in the margent the second shewes whether the division be right or no the Quotient is 413 the number of Bissextiles which now I make the Multiplicand and 4 which was the Divisor the Multiplier whose product is 1652 unto which adde the Remainder 2 in the Division the summe will be 1654 as before in the second Table of the margent is evident And thus is Division tried by Multiplication Example 13. By the third example of Multiplication Minutes 525960     Quotient 8766 Houres 8766 it was desired to know the number of minutes in a vulgar yeare containing 365 dayes and 6 houres 24 making a naturall day that is 8766 houres in a year which was there the Multiplicand and 60 the minutes in an houre was the Multiplier which here I make the Divisor and the Product 525960 minutes the Dividend which by the tenth example of Division may be divided by 6 cutting off a cypher upon the right hand of the Dividend and then the Quotient will be found 8766 houres the Multiplicand as before which proves the Multiplication true and so the like of any other if any thing had remained the last place cut off in the Dividend must have been restored unto it and the cypher likewise to the Divisor A Memorandum Observe in this last way of Division that how many points there be in the Dividend so many figures or cyphers there must be in the Quotient and that every remainder must be lesse then the Divisor otherwise the Quotient is too little or the operations wrong and when any number is given for to be divided if you can finde a number that will divide both the Dividend and Divisor without leaving any remainder they will remaine in the same proportion as when cyphers are cut off from either and if their Quotients doe divide one an other they will produce an other Quotient equall to the first and their remainder if there be any shall have still the same proportion as for example if 48 were to be divided by 12 the Quotient would be 4 and 't will be so if you take the halfe of these as 24 to be divided by 6 or 12 by 3 the Quotient will be 4 as you shall see more at large hereafter in all wayes of Division if the Dividend ends with an odde number in the place of unites viz. 1 3 5 7 9 and the Divisor even there must be a remainder when the division is done for any odde number Multiplied by one that is even the product will be even although the Quotient be odde all numbers may be divided into 2 equall parts if the figure be even in the unite place if odde it cannot without a fraction if any number hath 5 or 0 in the unite place 5 will divide it without any remainder but when any thing does remaine after Division is ended although it were a part of the Dividend yet as it hath relation to the Divisor it must be of the same denomination with the Quotient as it is a fraction or part of an integer as if 10 s. were to be divided betwixt 4 men the Quotient will be 2 and the remainder 2 s. but as it hath relation to four men it is but 6 d. or two fourths of a shilling the Quotients denomination and should be annext unto it as a fraction according to the next Section and Paragraphs instructions The Second SECTION treats of broken numbers or parts of integers divided into 5 Paragraphs demonstrating Reduction Addition Subtraction Multiplication and Divisions both of proper and improper Fractions Paragraph 1. Explicating the definition termes value and qualities of Fractions and how to reduce them from one denomination to another as Fractions to Integers and the contrary Definitions and Termes WHen the Dividend is lesse than the Divisor it is said to be a part of an Integer or the whole called a Fraction or broken number subscribed underneath one another with a line drawn between them in this manner ½ Fractions are proper improper or mixt The termes proper to Fractions are usually these the uppermost of the two is commonly called the Numerator of the Fraction the 1 Numerator 2 Denominator other Denominator and thus as in the margent Fractions demonstrated A Fraction being defined a broken number or part of any integer the