Selected quad for the lemma: end_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
end_n	line_n	point_n	require_v	1,303	5	9.1027	5	false

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A48262	Mathematicall recreations. Or, A collection of many problemes, extracted out of the ancient and modern philosophers as secrets and experiments in arithmetick, geometry, cosmographie, horologiographie, astronomie, navigation, musick, opticks, architecture, statick, mechanicks, chemistry, water-works, fire-works, &c. Not vulgarly manifest till now. Written first in Greeke and Latin, lately compi'ld in French, by Henry Van Etten, and now in English, with the examinations and augmentations of divers modern mathematicians whereunto is added the description and use of the generall horologicall ring: and the double horizontall diall. Invented and written by William Oughtred.; Récréation mathématique. English.	Oughtred, William, 1575-1660. aut	1653 (1653)	Wing L1790; ESTC R217635	140,825	339

a line stricken perpendicular upon it apply the streight edg unto the wall at what time the sun shineth upon it holding the board parallel to the horizon Set the dyal thereon and move it gently every way untill the same hour and minute be shewed in both dyals and so let it stand then if the dyal have one of the sides parallel to the Meridian strike a line along that side upon the board crossing the perpendicular or else with a bodkin make a point upon the board at each end of the meridian and taking away the instrument from the board and the board from the wall lay a ruler to those two points and draw a line crossing the perpendicular for the angle which that line maketh with the perpendicular is the angle of the decli●nation of the wall And if it be a right angle the wall is exactly east or west but if that line be parallel to the perpendicular the wall is direct north or south without any declination at all You may also finde out the declination of a wall if the dial be fixed on a post not very far from that wall in this manner Your board being applyed to the wall as was shewed hang up a thred with a plummet so that the shadow of the thred may upon the board crosse the perpendicular line make two pricks in the shadow and run instantly to the dyal and look the horizontal distance of the suns Azumith or upright shadow from the meridian Then through the two pricks draw a line crossing the perpendicular and upon the point of the intersection make a circle equal to the horizon of your Instrument in which Circle you shal from the line through the two pricks measure the Horizontal distance of the upright shadow or Azumith from the meridian that way toward which the Meridian is draw a line out of the center to the end of that arch measured and the angle which this last line maketh with the perpendicular shall be equall to the declination of the wall XIII Vse How to place the dyall upon a post without any other direction but it selfe Set the diall upon the post with the stile into the North as neere as you can guesse then move it this way and that way till the same houre and minute be shewed both in the outward and inward dials by the severall shadowes as hath been already taught for then the diall standeth in its truest situation wherefore let it be nailed down in that very place XIIII Vse To finde the height of the sun at high noon everyday Seeke out the diurnall Arch or parallel of the suns course for that day by Vse III. and with a paire of Compasses setting one foot in the center and the other in the point of intersection of that parallel with the Meridian apply that same distance unto the Semidiameter divided for that measure shal therein shew the degree of of the Suns altitude above the the Horizon that day at high noon XV Vse To finde the height of the sun at any houre or time of the day Seeke out the diurnal Arch or parallel of the suns course for that day and marke what point of it is in the very houre and minute proposed And with a paire of Compasses setting one foot in the Center and the other in that point of the parallel apply the same distance upon the Semidiameter divided for that measure shall shew the degree of the suns altitude above the Horizon at that time And by this meanes you may finde the height of the Sun above the Horizon at every houre throughout the whole yeere for the making of rings and cylinders and other instruments which are used to shew the houre of the day XVI Vse The height of the sun being given to finde out the houre or what it is a clocke This is the converse of the former Seeke therefore in the Semidiameter divided the height of the sun given And with a paire of Compasses setting one foot in the center and the other at that height apply the same distance unto the diurnall arch or parallel of the Sun for that day for that point of the diurnall arch upon which that same distance lights is the true place of the sun upon the dial and sheweth among the houre lines the true time of the day XVII Use. Considerations for the use of the instrument in the night In such questions as concerne the night or the time before sun rising and after sun setting the instrument representeth the lower Hemisphaere wherein the Southerne pole is elevated And therefore the parallels which are above the Aequinoctiall toward the center shall be for the Southerne or winter parallels and those beneath the Aequinoctiall for the Northerne or Summer paral●els and the East shall be accounted for West and the West for East altogether contrary to that which was before when the Instrument represented the upper Hemisphaere XVIII Use. To finde how many degrees the sun is under the Horizon at any time of the night Seeke the Declination of the sun for the day proposed by Vse II. And at the same declination the contrary side imagine a parallel for the sun that night and mark what point of it is in the very houre and minute proposed And with a pair of compasses setting one foot in the center and the other in that point of the parallel apply that same distance unto the semidiameter divided for that measure shall shew the degree of the suns depression below the Horizon at that time XIX Use. To finde out the length of the C●epusculum or twylight every day Seek the declination of the sun for the day proposed by Vse II. And at the same declination on the contrary side imagine a parallel for the sun that night And with a paire of compasses setting one foot in the center and the other at 72 degrees upon the semidiameter divided apply that same distance unto the suns nocturnall parallel for that point of the parallel upon which that same distance shall light sheweth among the houre lines the beginning of the twilight in the morning or the end of the twilight in the evening XX Use. If the day of the moneth be not known to finde it out by the dyall For the working of this question either the diall must be fixed rightly on a post or else you must have a true Meridian line drawn in some window where the sun shineth wherefore supposing the diall to be justly set either upon the post or upon the Meridian Look what a clock it is by the outward diall and observe what point of the upright shadow falleth upon the very same minute in the inner diall and through that same point imagine a parallel circle for the suns course that imaginary circle in the Ecliptick shall cut the day of the moneth I The description of it THis Instrument serveth as a Diall to finde the houre of the day not in one place onely as the most part of Dials do but generally in all Countreys lying North of the Aequinoctiall and therefore I call it the generall H●rologicall ●ing It consisteth of two br●zen circles a Diameter and a little Ring to hang it by The two circles are so made that though they are to be set at right angles when you use the Instrument yet for more convenient carrying they may be one folded into the other The lesser of the two circles is for the Aequinoctiall having in the midst of the inner side or thicknesse a line round it which is the true Aequinoctiall circle divided into twice twelue hours from the two opposite points in which it is fastened within the greater The greater and outer of the two circles is the Meridian One quarter whereof beginning at one of the points in which the Aequin●cti●ll is hung is divided into ninety degrees The Diameter is fastened to the Meridian in two opposite points or poles o●e of them being the very end of the Quadrant and is the North Pole Wherefore it is perpendicular to the ●quinoctiall having his due position The diameter is broad and slit in the middle and about the slit on both sides are the moneths and dayes of the yeer And within this slit is a litt●e sliding plate pierced through with a small hole which hole in the motion of it while it is applied to the dayes of the yeer representeth the Axis of the world The little Ring whereby the Instrument hangeth is made to slip up and down along the Quadrant that so by help of a little tooth annexed the Instrument may be rectified to any elevation of the Pole II. The use of it IN using this Instrument First the tooth of the little Ring must carefully be set to the height of the Pole in the Quadrant for the place wherein you are Secondly the hole of the sliding plate within the slit must be brought exactly unto the day of the moneth Thirdly the Aeqinoctiall is to be drawn out and by means of the two studs in the Meridian staying it it is to be set perpendicular thereto Fourthly Guesse as neer as you can at the houre and turn the hole of the little plate toward it Lastly Hold the Instrument up by the little Ring that it may hang freely with the North Pole thereof toward the North and move it gently this way and that way till the beams of the Sun-shining thorow that hole fall upon that middle line within the Aequinoctiall for there shall be the houre of the day And the Meridan of the Instrument shall hang directly North and South These Instrument all Dials are made in brasse by Elias Allen dwelling over against St. Clements Church without Temple Barre at the signe of the Horse-shooe neere Essex Gate FINIS

because the marke C is seen at D move the Musket to and fro untill it doth agree with the line of reflection MB which suppose at LI so shall it be truly placed and giving fire to the Musket it shall not faile to strike the said mark at C. PROBLEM LXXX How to make an Image to be seen hanging in the aire having his head downeward TAke two Glasses and place them at right Angles one unto the other as admit AB and CB of which admit CB Ho●izontall and let the eye be at H and the object or image to be DE so D will be reflected at F so to N so to HE then at G so to ● and then to H and by a double reflection ED will seeme in QR the highest point D in R and the point L in Q inversed as was said taking D for the head and E for the feet so it will be a man inversed which will seem to be flying in the aire if the Jmage had wings unto it and had secretly 〈◊〉 motion and if the Glasse were bigge enough to receive many reflexions it would deceive the sight the more by admiring the changing of colours that would be seen by that motion PROBLEM LXXXI How to make a company of representative Souldiers seeme to be a Regiment or how few in number may be multiplyed to seem to be many in number TO make the experiment upon men there must be prepared two great Glasses but in stead of it we will suppose two lesser as GH and FI one placed right against another perpendicular to the Horizon upon a plaine levell Table betvveene vvhich Glasses let there be ranged in Battalia-vvise upon the same Table a number of small men according to the square G H I F or in any other forme or posture hen may you evidently see hovv the said battel vvill be multiplyed and seem farre bigger in the appearance than it is in effect Corolarie BY this invention you may make a little Cabinet of foure foot long and tvvo foot large more or lesse vvhich being filled vvith Rockes or such like things or there being put into it Silver Gold Stones of luster Jewels c. and the walls of the said Cabinet being all covered or hung with plaine glasse these visibles will appeare manifoldly increased by reason of the multiplicitie of reflexions and at the opening of the said Cabinet having set something which might hide them from being seen those that look into it will be astonished to see so few in number which before seemed to be so many PROBLEM LXXXII Of fine and pleasant Dyal● COuld you choose a more ridiculous one than the natural Dyall written amongst the Greek Epigrams upon which some sound Poet made verses shewing that a man carrieth about him alwayes a Dyall in his face by meanes of the Nose and Teeth and is not this a jolly Dyall for he need not but open the mouth the lines shall be all the teeth and the nose shall serve for the style Of a Dyall of hearbes CAn you have a finer thing in a Garden or in the middle of a Compartemeet than to see the lines and the number of houres represented with little bushie hearbes as of Hysope or such which is proper to be cut in the borders and at the top of the style to have a Fanne to shew which way the winde b●oweth this is very pleasant and useful Of the Dyall upon the fingers and the hand IS it nor a commoditie very agreeable when one is in the fie●d or in some vil●age vvithout any other Dyall to see onely by the hand what of the clock it is vvhich gives it very neare and may be practised by the left hand in this manner Take a stravv or like thing of the length of the Index or the second finger hold this straw very right betvveen the thumb and the fore-finger then stretch forth the hand and turne your back and the palm of your hand tovvards the Sunne so that the shadovv of the muscle vvhich is under the Thumb touch the line of life vvhich is betvveen the middle of the tvvo other great lines vvhich is seen in the palme of the hand this done the end of the shadovv vvill shevv vvhat of the clock it is for at the end of the first finger it is 7 in the morning or 5 in the evening at the end of the Ring-finger it is 8 in the morning or 4 in the evening at the end of the little finger or first joynt it is 9 in the morning or 3 in the after-noone 10 2 at the second joynt 11 and 1 at the third joynt and midday in the line follovving vvhich comes from the end of the Index Of a Dyall which was about an Obeliske at Rome WAs not this a pretty fetch upon a pavement to choose an Obeliske for a Dyall having 106 foot in height without removing the Basis of it Plinie assures us in his 26 book and 8 Chap. that the Emperour Augustus having accom●odated in the field of Mars an Obeliske of this height he made about it a pavement and by the industry of Man●lius the Mathematician there were enchaced markes of Copper upon the Pavement and placed also an Apple of Gold upon the toppe of the said Obeliske to know the houre and the course of the Sunne with the increase and decrease of dayes by the same shadow and in the same manner do some by the shadow of their head or other style make the like experiments in Astronomie Of Dyals with Glasses PT●lomie w●ites as Cardanus reports that long ago there were Glasses which served for Dyals and presented the face of the beholder as many times as the houre ought to be twice if it were 2 of the clock 9 if it were 9 c. But this was thought to be done by the help of water and not by Glasses which did leake by little and little out of the vessell discovering anon one Glasse then anon two Glasses then 3 4 5 Glasses c. to shew so many faces as there were houres which was onely by leaking of water Of a Dyall which hath a Glasse in the place of the Style WHat will you say of the invention of Mathematicians which finde out daily so many fine and curious novelties they have now a way to make Dyals upon the wainscot or seeling of a Chamber and there where the Sunne can never shine or the beames of the Sunne cannot directly strike and this is done in placing of a little Glasse in the place of the style which reflecteth the light with the same condition that the shadow of the style sheweth the houre and it is easie to make experiment upon a common Dyall changing only the disposition of the Dyall and tying to the end of the style a piece of plaine Glasse The Almaines use it much who by this way have no greater trouble but to put their Noses out of their beds and see what a clock it is which is reflected

circuit of the earth is 21600 miles From the Centre of the earth to the Moone there is neare 56 Semidiameters of the earth which is about 192416 miles unto the Sunne there is 1142 Semidiameters of the earth that is in miles 3924912 from the starry firmament to the Centre of the earth there is 14000 Semidiameters that is 48184000 miles according to the opinion and observation of that learned Ticho Brahe From these measures one may collect by Arithmeticall supputations many pleasant propositions in this manner First if you imagine there were a hole through the earth and that a Milstone should be let fall down into this hole and to move a mile in each minute of time it would be more than two dayes and a halfe before it would come to the Centre and being there it would hang in the aire Secondly if a man should go every day 20 miles it would be three yeares wanting but a fortnight before he could go once about the earth and if a Bird should fly round about it in two dayes then must the motion be 450 miles in an houre Thirdly the Moone runnes a greater compasse each houre than if in the same time she should runne twice rhe Circumference of the whole earth Fourthly admit it be supposed that one should go 20 miles in ascending towards the heavens every day he should be above 15 years before he could attaine to the Orbe of the Moone Fifthly the Sunne makes a greater way in one day than the Moone doth in 20 dayes because that the Orbe of the Sunnes circumference is at the least 20 times greater than the Orbe of the Moone Sixthly if a Milstone should descend from the p●ace of the Sunne a thousand miles every houre which is above 15 miles in a minute farre beyond the proportion of motion it would be above 163 dayes before it would fall dovvne to the earth Seventhly the Sunne in his proper sphere moves more than seven thousand five hundred and seventy miles in one minute of time novv there is no Bullet of a Cannon Arrovv Thunderbolt or tempest of vvinde that moves vvith such quicknesse Eightly it is of a farre higher nature to consider the exceeding and unmoveable quicknesse of the starry firmament for a starre being in the Aequator which is just between the Poles of the world makes 12598666 miles in one houre which is two hundred nine thousand nine hundred and seventy foure miles in one minute of time if a Horseman should ride every day 40 miles he could not ride such a compasse in a thousand yeares as the starry firmament moves in one houre which is more than if one should move about the earth a thousand times in one houre and quicker than possible thought can be imagined and if a starre should flye in the aire about the earth with such a prodigious quicknesse it would burne and consume all the world here below Behold therefore how time passeth and death hasteth on this made Copernicus not unadvisedly to attribute this motion of Primum mobile to the earth and not to the starry firmament for it is beyond humane sense to apprehend or conceive the rapture and violence of that motion being quicker than thought and the word of God testifieth that the Lord made all things in number measure weight and time PROBLEM XCII To finde the Bissextile yeare the Dominicall letter and the letters of the moneth LEt 123 or 124 or 125 or 26 or 27 which is the remainder of 1500 or 1600 be divided by 4 which is the number of the Leape-yeare and that which remaines of the division shewes the leap-yeare as if one remaine it shewes that it is the first yeare since the Bissextile or Leap-year if two it is the second year c. and if nothing remaine then it is the Bissextile or Leap-yeare and the Quotient shews you how many Bissextiles or Leap-yeares there are conteined in so many yeares To finde the Circle of the Sun by the fingers LEt 123 24 25 26 or 27 be divided by 28 which is the Circle of the Sunne or whole revolution of the Dominicall letters and that which remaines is the number of joynts which is to be accounted upon the fingers by Filius esto Dei coelum bonus accipe gratis and where the number ends that finger it sheweth the yeare which is present and the words of the verse shew the Dominicall letter Example DIvide 123 by 28 for the yeare and so of other yeares and the Quotient is 4 and there remaineth 11 for which you must account 11 words Filius esto Dei c. upon the joynts beginning from the first joynt of the Index and you shall have the answer For the present to know the Dominicall letter for each moneth account from January unto the moneth required including January and if there be 8 9 7 or 5 you must begin upon the end of the finger from the thumbe and account Adam degebat c. as many words as there are moneths for then one shall have the letter which begins the moneth then to know what day of the moneth it is see how many times 7 is comprehended in the number of dayes and take the rest suppose 4 account upon the first finger within without by the joynts unto the number of 4 which ends at the end of the finger from whence it may be inferred that the day required was Wednesday Sunday being attributed to the first joynt of the first finger or Index and so you have the present yeare the Dominicall letter the letter which begins the Moneth and all the dayes of the Moneth PROBLEM XCIII To finde the New and Full Moone in each Moneth ADde to t●e Epact for the yeare the Moneth from March then subtract that surplus from 30 and the rest is the day of the Moneth that it vvill be New Moone and adding unto it 14 you shall have that Full Moone Note THat the Epact is made alwayes by adding 11 unto 30 and if it passe 30 subtract 30 and adde 11 to the remainder and so ad infinitum as if the Epact were 12 adde 11 to it makes 23 for the Epact next year to vvhich adde 11 makes 34 subtract 30 rests 4 the Epact for the yeare after and 15 for the yeare follovving that and 26 for the next and 7 for the next c. PROBLEM XCIV To finde the Latitude of ● Countrey THose that dwell between the North-Pole and the Tropicke of Cancer have their Spring and Summer between the 10 of March and the 13 of September and therefore in any day between that time get the sunnes distance by instrumentall observation from the zenith at noone and adde the declination of the sun for that day to it so the Aggragate sheweth such is the Latitude or Poles height of that Countrey Now the declination of the sunne for any day is found out by Tables calculated to that end or Mechanically by the Globe or by Instrument it may

Compasses so much is the distance between the two places If the distance of two places be required in a particular Map then with the Compasses take the distance between the two places and apply it to the scale of Miles so have you the distance if the scale be too short take the scale between the Compasses and apply that to the two places as often as you can so have you the distance required Of the Longitude Latitude Declination and distance of the Starres THe Declination of a starre is the nearest distance of a Star from the Aequator the Latitude of a Starre is the nearest distance of a Sarre from the Ecliptick the Longitude of a Starre is an Ark of the Ecliptick conteined between the beginning of Aries and the Circle of the Starres Latitude which is a circle drawne from the Pole of the Ecliptick unto the starre and so to the Ecliptick The distance between two Sarres in Heaven is taken by a Crosse-staffe or other Instrument and upon a Globe it is done by taking between the feet of the Compasses the two Starres and applying it to the Aequator so have you the distance betweene those two starre● How is it that two Horses or other creatures being foaled or brought forth into the world at one and the same time that after certaine dayes travell the one lived more dayes than the other notwithstanding they dyed together in one and the sam● moment also THis is easie to be answered let one of them travell toward the West and the other towards the East then that which goes towards the West followeth the Sunne and therefore shall have the day somewhat longer than if there had been no travell made and that which goes East by going against the Sunne shall have the day shorter and so in respect of travell though they dye at one and the selfe same houre and moment of time the one shall be older than the other From which consideration may be inferred that a Christian a Jew and a Saracen may have their Sabbaths all upon one and the same day though notwithstanding the Saracen holds his Sabath upon the Friday the Jew upon the Saturday and the Christian upon the Sunday For being all three resident in one place if the Saracen and the Christian begin their travell upon the Saturday the Christian going West and the Saracen Eastwards shall compasse the Globe of the earth thus the Christian at the conclusion shall gaine a day and the Saracen shall lose a day and so meet with the Jew every one upon his owne Sabbath Certaine fine observations 1 UNder the Equinoctiall the Needle hangs in equilibrio but in these parts it inclines under the Horizon and being under the Pole it is thought it will hang verticall 2 In these Countreys which are without the Tropicall Circles the Sunne comes East and West every day for a halfe yeare but being under the Equinoctiall the Sun is never East nor West but twice in the yeare to wit the 10. of March and the 13 of September 3 If a ship be in the Latitude of 23 gr 30 m. that is if it have either of the Tropicks verticall then at what time the Sunnes Altitude is equall to his distan●e from any of the Equinoctiall points then t●e Sunne is due East or West 4 If a ship be betweene the Equinoctiall and either of the Tropicks the Sunne will come twice to one point of the compasse in the forenoone that is in one and the same position 5 Vnder the Equinoctiall neare Guinea there is but two sorts of windes all the year 6 moneths a Northerly winde and 6 moneths a Southerly winde and the flux of the Sea is accordingly 6 If two ships under the Equinoctiall be 100. leagues asunder and should sayle Northerly untill they were come under the Articke circle they should then be but 50 leagues asunder 7 Those which have the Artick circle verticall when the Sunne is in the Tropick of Cancer the Sun setteth not but toucheth the western part of the Horizon 8 If the complement of the Sunnes height at noon be found equall to the Sunnes Declination for that day then the ●quinoctiall is verticall or a shippe making such an observation the Equinoctiall is in the Zenith or direct over them by which Navigators know when they crosse the line in their travels to the Indies or other parts 9 The Sunne being in the Equinoctiall the extremity of the stile in any Sunne-dyall upon a plaine maketh a right line otherwise it is Eclipticall Hyperbolicall c. 10 When the shadow of a man or other thing upon a Horizontall 〈◊〉 is equall unto it in length then is the Sunne in the middle point between the Horizon and the Zenith that is 45 degrees high PROBLEM XCVII To make a Triangle that shall have three right Angles OPen the C●passes at p●easure and upon A describe an Arke BC. then at the same opening place one of the feet in B and describe the Ark AC Lastly place one of the feet of the Compasses in C. and describe the Arke AB· so shall you have the sphericall Aequilaterall Triangle ABC right angled at A at B and at C. that is each angle comprehended 9● degrees which can never be in any plaine Triangle whether it be Equilaterall Isocelse scaleve Orthogonall or Opigonall PROBLEM XCVIII To divide a line in as many equall parts as one will without compasses or without seeing of it THis Proposition hath a fallacie in it cannot be practised but upon a Maincordion for the Mathematicall line which proceeds from the flux of a point cannot be divided in that wise One may have therefore an Instrument which is called Maincordion because there is but one cord and if you desire to divide your line into 3 parts run your finger upon the frets untill you sound a third in musick if you would have the fourth part of the line then finde the fourth sound a fifth c. so shall you have the answer PROBLEM XCIX To draw a line which shall incline to another line yet never meet against the Axiome of Parallels THis is done by help of a Conoeide line produced by a right line upon one the same plaine held in great account amongst the Ancients and it is drawne after this manner Draw a right line infinitely and upon some end of it as at I draw a perpendicular line I A. augment it to H. then from A. draw lines at pleasure to intersect the line I. M. in each of which lines from the right line IM transferre IH viz. KB LC.OD.PE.QF.MG then from those points draw the line H.B.C.D.E.F.G. which will not meet with the line IM and yet incline nearer and nearer unto it PROBLEM C. To observe the variation of the compasses or needle in any places FIrst describe a Circle upon a plaine so that the Sun may shine on it both before noone and afternoone in the centre of which Circle place a Gn●●on or wire perpendicular