Selected quad for the lemma: book_n

Word A	Word B	Word C	Word D	Occurrence	Frequency Band	MI	MI Band	Prominent
book_n	construction_n	demonstration_n	part_n	4,914	5	10.2330	5	true

Text snippets containing the quad

ID	Title	Author	Corrected Date of Publication (TCP Date of Publication)	STC	Words	Pages
A33999	Geometrical dyalling, or, Dyalling performed by a line of chords onely, or by the plain scale wherein is contained two several methods of inscribing the hour-lines in all plains, with the substile, stile and meridian, in their proper coasts and quantities : being a full explication and demonstration of divers difficulties in the works of learned Mr. Samuel Foster deceased ... : whereto is added four new methods of calculation, for finding the requisites in all leaning plains ... : also how by projecting the sphere, to measure off all the arks found by calculation ... : lastly, the making of dyals from three shadows of a gnomon ... / written by John Collins ...	Collins, John, 1625-1683.	1659 (1659)	Wing C5373; ESTC R17003	56,110	109

First therefore by the former directions prick off the Substile Stile and Meridian in their true Coast and quantity and perpendicular to the Meridian draw a line passing through the Center and A South Diall Declin 40d East Inclin 15d Lat 51d 32′ A North Diall Declin 40d East Reclining 75d Lat 51d 32′ it shall represent the Horizontal line of the Plain in that new Latitude as here VS from any point in the Stile as K let fall a perpendicular to the Substile at I and from the point I in the Substile let fall a perpendicular to the Meridian at P. To finde the new Declination Prick IP on the substilar line from I to R and draw RK so shall the Angle IRK be the complement of the new declination and the Angle IKR the new Declination it self To finde the new Latitude Upon the Center V with the Radius VK describe a Circle I say then that VP is the Sine of the new Latitude to that Radius which may be measured in the Limbe of the said Circle by a line drawn parallel to VS which will intersect the Circle at F so is the Arch SF the measure of the new Latitude To prick off the Hour-line of Six This must be prickt off below the Horizontal line the same way that the substilar lyes The proportion is As the Cotangent of the Latitude Is to the Sine of the Declination So is the Radius To the Tangent of the Angle between the Horizon and six If RK be Radius then is VP the Tangent of the new Latitude but if we make VP Radius then is RK the Cotangent of the new Latitude Wherefore prick the extent RK on the Horizontal line from V to N and thereon erect the Sine of the Declination to the same Radius perpendicularly as is NA and a line drawn into the Center shall be the hour-line of six the proportioning out of the Sine of the Declination to the same Radius will be easily done enter the Radius VP from K to D and the nearest distance from D to IK shall be the Sine of the new declination to that Radius To fit in the Parallelogram This is to be done as in upright Decliners for having drawn a line from the new Latitude at F into the Center if any Radius be entred on the said line from V the Center towards the Limbe the nearest distance from that point to VP the Meridian line shall be the Cosine of the new Latitude to that Radius Again if the same Radius be entred on RK produced if need be the nearest distance to VR produced when need requires shall be the Cosine of the new Declination and then the hour-lines are to be drawn as for upright decliners nothing will be doubted concerning the truth of what is here delivered if the demonstration for inscribing the Requisites in upright decliners be well understood it being granted that Oblique Plains in some Latitude or other will become upright decliners There are two Examples for the Latitude of London suited to these directions in both which the Letters are alike the one for a South Plain declining 40d Eastwards Inclining 15d the other for a North Plain declining 40d East Reclining 75d To finde a true Meridian Line For the true placing of an Horizontal Dyal as also for other good uses it will be requisite to draw a true Meridian Line which proposition may be performed several ways amongst others the Learned Mathematician Francis van Schooten in his late Miscellanies demonstrates one performed by help of three shadows of an upright Stile on a Horizontal Plain published first without Demonstration in an Italian book of dyalling by Mutio Oddi But if all three be unequal as let AC be the least erect three lines from the point A perpendicular to AP AC AD as is AF AG and AH equal to the Stiles height AE and draw lines from the extreamities of the three shadows to these three points as are FB GC and HD then because AC is less then AB therefore GC will be less then FB by the like reason GC will be less then HD wherefore from FB and HD cut off or Substract FI and HK equal to GC and from the points I and K let fall the perpendiculars IL KM upon the Bases AB AD afterwards draw a line joyning the two points M L and from the said points let fall the perpendiculars LN equal to LI and MO equal to MK Then because the two shadows AB and AD are unequal in like manner FB and HD will be unequal but forasmuch as FI and HK are equal by construction it follows that LI KM or LN and MO will be unequal and forasmuch as these latter lines are parallel a right line that connects the points O and N being produced will meet with the right line that joyns M L produced as let them meet in the point P from whence draw a Line to C and it shall be a true line of East and West and any Line perpendicular thereto shall be a Meridian line thus the perpendicular AQ let fall thereon is a true Meridian Line passing through the point A the place of the stile or wyre Whereto I adde that if MO LN retaining their due quantities be made parallel it matters not whether they are Perpendicular to ML or no also for the more exact finding the Point P the lines MO and LN or any other line drawn parallel to them may be multipyled or increased both of them the like number of times from the points M and L upwards as also from the points O and N downwards and Lines drawn through the points thus discovered shal meet at P without producing either ML or ON See 15 Prop. of 5 Euclid and the fourth of the sixth Book The greater part of van Schootens Demonstration is spent in proving that ML and ON produced will meet somewhere this for the reasons delivered in the construction I shall assume as granted then understand that the three Triangles ABF ACG and ADH stand perpendicularly erect on the plain of the Horizon beneath them upon the right Lines AB AC and AD whence it will come to pass that the three points F G and H meet in one point as in E the top of the Stile AE and that the right lines FB GC and HD are in the Conique Surface of the shadow which the Sun describes the same day by his motion the top of which Cone being the point E. Wherefore if from those right Lines we substract or cut off the right Lines FI GC and HK being each of them equal to one another then will the points I C and K fall in the circumference of a Circle the plain whereof is parallel to the plain of the Equator and therefore if through the points K and I such a Position a right Line be imagined to pass and be produced to the plain of the Horizon it will meet with ML produced in the point P