A Plane.

BY a Plane is meant any thing which is flat or smooth, bee it Wall, board, Stone, or any thing els whereon lines may bee conveniently drawn; which must bee conceived to have length and bredth, but no thickness; that so when the Sun forsake's the one face or side of any Plane, it must presently appear upon the other:for every Plane hath two faces or sides, whereon som hour-lines may bee drawn for good use; Except the Horizontal, which can have but one.

Again, everie Plane must cut or divide the Globe of the Univers into two equal halvs:As the Horizontal Plane divide's the upper Hemisphere which wee see, from the lower which wee see not:Or the Equinoctial Plane may bee said to divide the Northern half of the Globe from the Southern half thereof, or the Meridian Plane; to divide the Eastern half from the Western half thereof:and so do all other Planes whatsoever, according to their several situations.

The Horizontal line.

THe Horizontal line, is a line alwaies level, or paralel to the Ho­rizon; beeing indeed a level-line.

A Perpendicular line.

A Perpendicular line is a line crossing the Horizontal, or any other line at right angles, but most proper, that line which crosseth the Horizontal at right angles, pointing out the Zenith and Na­der.

Declination.

THe Declination of a Plane is, what a Plane varie's or differ's from beholding the due North or South, either to the East or Westwards.

Inclination.

THe Inclination of a Plane is the quantitie of an Arch in degrees and minutes which any plane is seated, or lifted up above the le­vel of the Horizontal line.

Pole of the world.

THe Pole of the world is represented in the Instrument by the letter P, through which all the Meridians pass.

Pole of the Plane.

THe Pole of the Plane must ever bee 90:00″ from the Plane it self, even as the Pole of the world is 90:00″ from the Equator.

Pole of the Meridian of the Plane.

SO must also the Pole of the Meridian of the Plane bee 90:00″ from the Meridian it self; which will bee far better understood in every of the several Diagrams following.

Meridian or Hour-circle.

BY a Meridian is meant any of the Hour-circles in the Instrument, whereof there are but 12 drawn, thereby pointing out the 24 hours, beeing 15:00″ each from other, which is an hour of time, and are somtimes termed Meridians, and somtimes Hour-circles, as occasion shall bee given.

The proper Meridian of the Plane.

THe proper Meridian of the Plane is that very Meridian, either real in the Instrument, or prickt down, which passeth through the Pole of the Plane.

Meridian of the place.

THe Meridian of the Place is that Meridian in this our Horizon, which make's (the Sun beeing thereupon) Mid-daie and Mid­night with us.

The Prime Vertical Circle.

THe Prime Vertical Circle is the Circle or Azumith which pas­seth through the Zenith and the points of East and West in the Instrument, represented by the line W z E.

The Scale.

THe Scale of the Instrument is the Radius or Semidiameter there­of, viz. z S beeing divided into 90:00 unequal degrees.

The limb of the Instrument.

THe limb of the Instrument is often named, by which you are to understand is meant the innermost Circle of all in the Instrument, which is in every of the Diagrams or Schemes divided into 360:00″.

An Hour-line.

BY an Hour-line is meant any streight line drawn from the Pole of the Plane, through the Intersection or crossing of any hour­circle with the Plane unto the limb of the Instrument.

The Substiler line.

THe Substiler line is likewise a streight line drawn from the Pole of the Plane, through the intersection or crossing of the proper Meridian of the Plane, with the Plane it self, unto the limb of the In­strument, over which line the Stile or Gnomon is alwaies plac'd.

Diagram.

BY the word Diagram is meant a Geometrical figure, whereof for every proposed plane (in the ensuing Tract) there are two of the same number from n^o A to ndeg; K, (both included).

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An explanation of the Instrument.

THis Instrument is our Hemisphere in Plane for the Latitude 51:30 first projected by M^r William Oughtred, and by him (as also by M^r Gunter) a Method is set down for the making thereof, wherein the Circle

• N E S W is the Horizontal Circle of the Plane.
• N z S the Meridian, or North and South line.
• W z E the Prime Vertical Circle, or East and West line.
• z represent's the Zenith, and somtimes the center of the Instru­ment.
• P the Pole of the world.
• W ♈ E the Equator.
• The Arch ♋ represent's the Summer Tropick.
• The Arch ♑ represent's the Winter Tropick.
• z S the Semidiameter is divided into 90:00″ unequal degrees.

Next there are drawn through the Pole in the the Instrumen P 12 Meridians which divide the whole face of the Plane into 24 parts, which are 15:00″ each from other (beeing an hour of time) there­fore they are taken and deemed for hour-circles also, and figured ac­cordingly, which will bee sufficient for our present matter. These several Diagrams following, beeing made onely for demonstration­sake; But the Instrument would bee much more useful and exact if it were larger, and every Meridian, or every 5^th Meridian of the 360:00″ were therein drawn, as hereafter will appear.

Further you are to note, that this Instrument is most proper for all dialls with centers, in which the hour-lines, substiler and stile do meet, whereof their natures, qualities and number follow:

First, The Horizontal Plane.

THe Instrument it self is an Horizontal diall:for if streight lines bee drawn from the center z to the several intersections of the hour-circles with the limb (which represent's the Horizontal Plane) and in z erect a Stile or Gnomon 51:30″ above the Plane, equal to N P right over the Meridian z P n It shall bee a perfect Horizontal Diall.

An upright Plane fully beholding the North or South.

THis Plane is likewise represented by the line W z E in the Instru­ment, beeing the Prime Vertical Circle, or East and West line thereof, beeing right to the Meridian N z S, therefore to the South­wards a due South face, and to the Northwards a due North face.

The Vertical Inclining Plane.

ALL Planes which are not level to, or with the Horizon, nor yet perpendicular thereto, but lie sloping betwixt the Zenith and Horizon, are called Inclining Planes:And if their Horizontal lines shall bee paralel to the Prime Vertical W z E, that is, fully beholding the North or South, they are then called Vertical Inclining Planes:These Planes may Incline either to the North-part of the Horizon, or to the South; and each Plane hath two faces whereon som hour­lines may bee drawn for use; The upper face lying open to the Ze­nith, and the lower to the Nader, wherein you are to consider the height of the Pole above the Plane by comparing the Inclination of the Plane to the Horizon, with the Latitude of the place:and will bee better understood by the demonstration following, wherein A c A represent the Horizontal line.

• z c n the Perpendicular line.
• N S the two Poles of the world, whereof N is elevated 51:30″ above A according to our Latitude.
• Ae c Ae the Equinoctial line or Plane.

The other lines are several Planes proposed according to their se­veral Inclinations to the Horizon.

I will first propose a Plane whose Inclination is 20:00″ in the North part of the Horizon, viz. the line B c B; now hereby you may plainly perceiv that the North-pole N is elevated above the upper face of this Plane but 31:30″; for if I take the angle of the Plane's Inclination A c B 20:00″ out of A c N 51:30″, there will rest the angle B c N 31:30″ for the North-Pole's height above the upper face, and therefore the South-pole S must bee elevated so much above the lower face of the said Plane as the angle B c S shew's, beeing the same with the angle B c N.

If the Plane proposed were D c D, Northerly likewise, you may [Page 7] perceiv by the situation of the Plane that the North-pole N will bee elevated above the lower face of this plane 20:00″; for if I take the angle A c N 51:30″ out of A c D 71:30″ there will rest the angle N c D 20:00″ for the North-pole's height above the lower face, and and therefore so much for the South-pole above the upper face there­of as the angle D c S shew's.

Again, if the Plane proposed were F c F 18:30 Inclination in the Southern part of the Horizon. Here, be­caus F in the South­part of the Horizon is lifted up above the Horizontal line 18:30″ according to the In­clination thereof; ther­fore F beeing the o­ther end of the Plane, must bee so much de­prest under it, so that if I add the angle AcF 18:30″ to the angle A c N 51:30″, I shall have the angle F c N 70:00″ for the North­pole's height above the upper face, and so much must the South-pole's height bee above the lower face of this Plane, as the angles F c N and F c S shew.

If the Plane proposed were the line N c S, it must of necessitie bee a Polar Plane, becaus it is the same with the Axis of the world, so that neither Pole will bee elevated above this Plane.

If the Plane proposed were the line Ae c Ae it must needs bee an Equinoctial Plane, and so the Poles will bee elevated 90:00″ above either face of the said Plane.

The upright declining Plane.

THese Planes, becaus they do not directly behold the South or North, nor yet the East or West fully, but are Planes which lie betwixt both, (and upright withall) are called upright declining Planes, which have likewise two faces whereon som hour-lines for use may bee drawn, the one face lying open to the Southwards, the other face to the Northwards.

The Declining Inclining Plane.

IF a Plane shall Decline anie waie from the due North or South ei­ther to the East or Westwards, and is not withall an upright plane, but is such a plane as with its Declination inclineth to the Horizon, it is then called a Declining Inclining Plane.

The Meridian Plane.

THose Planes whose Horizontal lines are the same, or parallel to the Meridian, and withall are upright Planes, are generally called Meridian Planes.

The Meridian Inclining Plane.

BUt those Planes whose Horizontal lines are the same, or parallel to the Meridian, and withall the Plane shall Incline to the Hori­zon, then it is called a Meridian Inclining Plane.

The Polar Plane.

THose Planes wherein a line may bee drawn paralel to the Axis of the world are called Polar Planes (although with som distincti­on) becaus that line pointeth unto the two Poles; and these Planes are alwaies paralel to som one of the hour-circles in the Instrument; For the direct Polar Plane is paralel to the hour-circle of 6, and the Meridian plane is paralel to the hour-circle of 12.

The Polar declining Plane.

BUt the Polar declining Plane is a Plane which is paralel to som one of the hour-circles in the Instrument betwixt 12 and 6, becaus of its Inclination to the Pole, and Declination from the [Page 9] prime Vertical, or, East and West line in the Instrument.

Next the Substiler and hour-lines in a Meridian Plane make an Arch with the Horizontal line of the plane, equal to the Latitude of the place:In direct Polar planes they make an Arch of 90:00″, but in Polar declining planes they make an Arch with their Horizontal line of their planes more then the latitude of the place, and less then 90:00″.

Again, in direct Polar planes the Substiler line is the same with the Meridian or 12 a clock line:In a Meridian plane it is the same with the hour of 6, but in a Polar declining plane it is betwixt 12 and 6, according to the inclination or distance of the proper Meri­dian of the plane with or from the Meridian of the place.

I should now proceed to the setting down those several planes up­on the Instrument in their order; but becaus som of them are plain regular dials, falling under every man's apprehension, and som com­prehended in what shall bee spoken of the others, I will omit these several following.

The first therefore which I shall omit or pass over, is the Hori­zontal plane, whereof the Instrument itself is a perfect pattern, as in folio 5 is declared.

The next following is the due South and North plane, which is likewise represented in the Instrument by the line W z E, and is fully comprehended and taught in any of the Diagrams no A B or C, of the upright declining planes.

The next following is the Vertical inclining plane, which is like­wise sufficiently explained in any of the Diagrams of number D E F G, of the declining inclining planes, there beeing but the same me­thod and order of working for the one as for the other.

The last which I pass over are the direct Polar and Equinoctial planes, beeing very plain regular Dialls, and indeed, not so proper for the Instrument as aforesaid.

First, The upright declining Plane.

THe first therefore which I will begin withall is, how to set down upon the Instrument any upright, or upright declining Plane:First you must bee very careful which waie your declination tend's; for if you mistake herein, your work must bee extreme fals.

This upright declining Plane is proposed to decline 30:00″ to the Eastwards of the South; therefore I reckon in the limb of the In­strument 30:00″ to the Northwards of E, and 30:00 to the South­wards of W, in the greater Diagram, number A, and so draw the line A z A for the Horizontal line of the Plane. Next I cross it at right angles with the line C z E for the perpendicular line thereof; for you must know that the Horizontal line of the Plane, and the Perpendicular line thereto, in all upright planes must bee the two first lines:and in all incliners the Horizontal line must bee the first, and a line crossing the Horizontal at right angles must bee the second (which upon the matter is all one) before you can proceed any fur­ther.

This beeing don, you perceiv the Southern face of this plane to behold the South-east quarter of the Instrument, and the Northern face thereof to behold the North-west quarter of the Instrument, which is according to the declination proposed.

How to set upon the Instrument any Vertical Incliner, or any Declining Inclining Plane.

FOr all sorts of Incliners you must have recours unto the Semidi­ameter or graduated scale in the Instrument z S.

Now this Plane Decline 24:20″ Westerly from S. is proposed to Incline 36:00″ Northerly.

First, becaus this plane decline's to the Southwestwarde, therefore in the greater Diagram number D I reckon from W in the limb 24:20″ Northerly, and so much from E Southerlie, and so I draw the Horizontal line of the Plane A z A, which I cross at right angles with the line B z B.

[Page 11]Next, I consider the Inclination proposed 36:00″ northerlie, which I take out of the Scale from S to e and put it over in the northern part of the line B z B from B to T, or, (which is much the better waie) I take the complement to 36:00″ beeing 54:00″ out of the Scale from z to e and put it over in the Northern part of the line from z to T, becaus the Inclination proposed is northerlie, of which you must have asmuch care as to the Declination; which point T will bee the highest point in this plane, beeing nearest the Zenith z. Now from the two ends of the Horizontal line A and A, and through this point T; having found a center, I draw the Arch of a circle A T A, which Arch is the Declining Inclining plane proposed:And these two foregoing examples will bee sufficient for setting down or pla­cing upon the Instrument all Planes whatsoëver.

To finde the place of the Pole of every upright Plane.

NOw after your Plane is justly set upon your Instrument with its horizontal and perpendicular lines, the next thing requisite to know, is the place of the Pole of your Plane, which must bee care­fully placed; for from the Pole of every Plane (having a center) the Substyler and all the hour-lines must be drawn to the limb of the Instrument.

You are likewise to know, that the Pole of every Plane is alwaies 90:00″ from the Plane it self, and ever, in all upright planes, at that end of the perpendicular line in the limb of the Instrument, to which the face of your Plane is opposite, for which you intend your Diall, becaus it is the Semidiameter, and therefore the Radius or Sine of 90:00″ which is general.

Thus in the greater Diagram no A the Southeasterlie face of the plane being proposed, therefore at the Southern end of the perpendi­cular line, viz. at C in the limb is the place of the Pole of that face of the plane. If the Northwesterlie face (which is the contrarie face) were intended for principal, then the Pole thereof would bee at E in the Northern end thereof.

To finde the place of the Pole of every Inclining, or of every Declining Inclining Plane.

IF the Plane bee a Vertical Incliner, or a Declining inclining Plane, as in the greater Diagram no D is proposed, then the Pole of the Plane must bee alwaies in the line which crosseth the Horizontal at right angles, betwixt the limb and the center z, so that you know z T beeing the complement of Inclination 54:00″ if I take 36:00″ out of the Scale from z downwards, it will extend to d, which bee­ing put over in the same line B z B from z, it will reach unto m, which must bee the place of the Pole of this Plane, becaus z T is 54:00″ of the Scale from z, and z m is 36:00″ from z, which make's the line T z m to bee 90:00″ betwixt the Plane at T, and the Pole thereof at m; and this is general for all Planes of this nature.

To draw the Hour-lines upon all Planes with Centers.

YOu are to mark diligently where the Hour-circles in the Instru­ment do cross or cut your Plane; for if a streight Ruler bee laid to the Pole of the Plane, and to those several intersections or cros­sings of the Hour-circles, with the Plane and streight lines drawn from those several intersections or crossings to the limb of the In­strument, those streight lines shall bee the Hour-lines for your Plane, and by reckoning the degrees in the limb of the Instrument you shall know, how much every hour-line is distant either from the Horizontal line of the Plane, or from the Perpendicular line there­of, or from the Substiler, or any one hour-line from another; by which you may transfer the same lines upon your Plane by a table made thereby as you pleas.

To draw the Substiler line upon all Planes with Centers.

YOu are to mark diligently where the Pole of your Plane fall's to bee either upon an hour-circle, or meridian, or betwixt any two; for if the Pole of the Plane fall upon a Meridian, as in the Diagram number D it doth fall just upon the Meridian or hour-circles of one [Page 13] in the point m, then this hour-circle of one must bee the proper Meridian of the Plane, becaus it passeth through the Pole of the Plane at m. Now if you laie a ruler to the Pole of the Plane m, and to the intersection or crossing of this Meridian with the plane it self, in the point L, and draw the streight line m L K to the limb of the Instrument, this line shall bee the Substiler line.

Furthermore, becaus the proper Meridian of the Plane, and the hour-circle of one, in this Diagram no D, bee one and the same, therefore the Substiler and hour-line of one, must bee one and the same line also, and serve for both.

But if the Pole of the Plane fall betwixt two meridians or hour­circles in the Instrument, as in the Diagram no A it doth, viz. be­twixt the hour circles of 9 and 10 in the Instrument, at the point C: in the limb (becaus it is an upright Plane) then for better demon­stration sake, I will in the Diagrams following, (where need shall re­quire) make or prick one down, which in this Diagram no A is C L P E, which is the proper meridian of this Plane, becaus it pas­seth through C the Pole thereof. Now laying a ruler to the Pole C, and to the Intersection of this proper meridian with the Plane it self in the point L, and draw the line C L K, it must bee the Substiler line for this Plane.

To finde the height of the Pole above all Planes having Centers.

THe next and last thing to know, is, how much the Pole is eleva­ted above your Plane:Now the height of the Pole above any Plane, is the measure of the arch of the proper meridian of the Plane, which is intercepted or included betwixt the Pole in the Instrument and the Plane:To know the quantitie of which arch in degrees and minutes, first reckon in the limb of the Instrument, from either end of the proper meridian 90:00″, mark where this acount end's, and from thence through the center z draw an obscure, or occult line to the opposite point in the limb, then take the distance betwixt the proper meridian and the center z in the occult line, and see how ma­ny degrees and minutes it is in the Scale from z downward; for the complement thereof beeing taken out of the Scale from z, and ex­tended in the occult line on the other side the center, shall point out [Page 14] the pole of this proper meridian; which had, laie a ruler to the Pole of this proper meridian, and to the Pole in the Instrument P, and make a mark in the limb where the ruler cut's. Again lay the ruler to the Poles of the proper meridian, and to the intersection or cros­sing of this meridian with the Plane, and make a second mark in the limb where the ruler cut's; for the degrees and minutes reckoned in the limb betwixt these two marks, will bee the exact measure of the arch of the proper meridian of the plane, which is intercepted or in­cluded betwixt the Pole and the Plane, which is the height of the Pole above all planes having centers.

Thus in the Diagram no A, from either end of the proper meridi­an of the Plane, viz. from E or C in the limb, I reckon 90:00″ which end's at A or A, beeing in this Diagram the ends of the horizantal line, and therefore the same line A z A serv's for the occult line be­fore spoken of (and will bee alwaies so in all upright planes). Next I finde the distance betwixt the proper meridian and the center z, viz. z L to bee in the Scale 21:00″; so that taking 69:00″ the comple­ment thereof out of the Scale from z downwards, and extending it in the occult line on the other side the center, it will point out the place of the Pole of this proper meridian to bee at h.

Now laying a ruler to the Pole of this proper meridian at h, and to the Pole in the Instrument P make a mark in the limb where the ruler cut's, viz. n; or (which is all one) draw the obscure line h P n; then keeping the ruler at h, turn it to the point L, where the proper Meridian cut's the Plane, and make the mark o in the limb, or the line h L o, and you shall finde the degrees and minutes betwixt n and o in the limb to bee 32:40″, which is the measure of the arch of the proper meridian betwixt P and L, which is the North-pole's height above the Northern face of the Plane, and therefore the South-pole must bee elevated so much above the Southern face thereof.

Again, in the Diagram no D, the proper meridian of the Plane which passeth through the Pole of the Plane, is the meridian or hour-circle of one, viz q m i L q: from q therefore, either end of this hour-circle, which is (now the proper meridian of the Plane) I rec­kon 90:00 in the limb, which end's at D, from which point D through the center z I draw the occult line D z D; Next I finde the distance betwixt the center z and the proper meridian in the obscure [Page 15] line to bee z i 9:00″ of the Scale; therefore taking the complement thereof 81:00″ out of the Scale from z downwards, it will extend on the other side the center, in the occult line from z to h, which will bee the place of the Pole of this proper meridian; now laying a ruler to h the Pole of the proper meridian, and to P the Pole in the Instrument, draw the obscure line h P n, then laying the ruler to h, and to the point L, where the proper meridian crosseth the Plane, and draw the line h L o, and you shall finde the degrees and minutes be­twixt n and o in the limb to bee 17:00″, which is the measure of the arch of the proper meridian betwixt P and L, which is the North­pole's height above the upper face of this declining inclining Plane, and therefore the South-pole must bee elevated so much above the lower face thereof.

To place the Stile or Gnomon.

NOw the height of the Pole above the Plane beeing found for right placing the Stile or Gnomon you must know it must ever point directly to the pole, which is elevated above that face of your plane for which you intend your Diall.

To know what hour-lines must bee in use for either face of any Plane whatsoever.

BEcaus you shall not incumber nor pester your Diall with useless or unnecessarie hour-lines, you shall most plainly perceiv, after your Plane is truly plac'd upon your Instrument, what hour-lines will bee in use for either face thereof, onely by observing the inter­sections, or crossings of the Tropiques and Parallels of the Sun's De­clination with the Plane, which is very plain and of excellent use.

For example in the Diagram no A▪ you perceiv the Southern face of the Plane to be A C A in the Instrument, and the Northern face there­of to bee A E A; now imagining the Sun's diurnal arch to bee in the Summer Tropique, you may perceiv that from its rising, which is about ¼ before 4 till it com to the point d, where the Tropique cros­seth the plane; it will shine upon the Northern face thereof, which is until about 40″ past 4 in the morn, from this point d: it immedi­ately [Page 16] appear's upon the Southern face thereof, and so continue's till the same Tropique cut's the Plane again in the point c, which you see will bee amongst the hour-circles about ½ an hour past 2, and then forsaking the Southern face, it appear's again upon the Northern face, and so continue's till Sun-set, which is about ¼ past 8 in the Evening.

So that for the Northern face, the hour-lines in use will bee first 4 in the morn, and then from 3 to 8, (both included); And for the Southern face, all the hour-lines from 5 in the morn, to 4 afternoon, (both included); becaus you see that when the Sun is in the parallel of 20:00″ South Declination, the Sun will shine upon that Southern face from its rising till its setting, beeing about a little past 4 after­noon, becaus that parallel 8, 20, 4, crosseth not the plane at all.

Again, in Diagram n^o D, the Sun beeing in the Summer Tropique, it will shine upon the Northern or lower face of that plane, from its rising to the crossing of that Tropique, with the Plane in the point a, which is about 25 past 6 in the morn, from thence in the same mo­ment of time, it will appear upon the Southern or upper face there­of, and continue's thereupon till it cross the Plane again in the point b. which is about 30″ past 7 in in the Evening, and then departing from that face, shine's upon the Northern face of the said Plane till Sun-set; But when the Sun is in 10:00″ South Declination, viz. the parallel g, i, 10, f, it will shine upon the Northern face from its rising at g, which is a little before 7 in the morn, till it cross the Plane in the point i, which is about ¼ past 7 in the morn; So that for the Northern, or lower face of the Plane, the hour-lines in use will bee from 4 to 7 in the morn, (both included) and onely 8 in the evening; and for the Southern or upper face, all the hour-lines from 7 in the morn to 7 at evening, both included.

Further you must know, that by the Instrument you shall alwaies have 12 of the 24 hour-lines pointed out unto you by the intersecti­ons or crossings of the hour-circles with the Plane.

It will fall out in all Southerlie Inclining, or in Southerlie Incli­ning Declining Planes, that although the Northern face of a Plane bee the face proposed to finde out the hour-lines and Substiler for, yet in these planes the hour-lines will bee drawn in and by the In­strument; for the lower or Southerlie face thereof, which is the [Page 17] contrarie face to the intended, which you shall fully understand in the Diagrams n^o E & G, following.

You are to note, that the Tables following for every Diagram, are the several distances in degrees and minutes taken out of the limb of the Instrument, which the Substiler and hour-lines are distant from the Perpendicular line of the plane in all upright Planes, and in all Incliners, from the line crossing the Horizontal at right angles, (which upon the matter is all one line).

Except the two Polar Declining Planes, whose hour-lines in the Tables take their several distances from the Substiler lines.

Lastly, in the 10 lesser Diagrams you have three whole circles, or rundles, in each whereof the greater (which represent's the Plane) is divided into 24 hour-lines by the Instrument, in which you have two scroles, whereof the outermost is the Diall for the face proposed, and in the innermost scrole are conteined the hour-lines which are in use for the contrarie face; and the two lesser circles or rundles are the Dialls for both faces of the proposed planes, where­of, in all Inclining planes, the rundle N^o V is for the upper, and the rundle N^o L stand's for the lower face of any of them.

Diagrams N^o A & A.

BEeing the first example of an Upright Declining Plane, propo­sed to decline 30:00″ Easterlie from South.

According to what hath been formerly taught I draw the line

• A z A for the Horizontal lines of this Plane.
• C z E for the Perpendicular lines of this Plane.
• C the Pole of the Plane becaus a Southerlie face is proposed.
• C L P E the proper Meridian of the Plane.
• L the point of this Meridian's cutting or crossing the Plane.
• C L K the Substiler line of this Plane.
• A z A serv's for the occult line, beeing 90:00 from C or E.
• z L 21:00″ the Meridian's distance from z in the occult line.
• z h 69:00 the Pole of this proper Meridian at h.
• n o in the limb, beeing the measure of the arch of this proper Me­ridian, betwixt P and L is 32:40″.

The Pole's height above this Plane 32:40″.

The hour-lines which are in use for the Northerlie face of this Plane, are four in the morn, and all the hour-lines from 3 to 8 (both included).

And for the Southern face thereof, all the hour-lines from 5 in the morn to 4 afternoon, both included.

Here in this Diagram n^o A, becaus the South-easterly face of the Plane is the face proposed to make a Diall for, therefore the hour­lines and substiler, as they are drawn in the semicircle of the Instru­ment A E A shall bee the perfect Diall, (without alteration) to bee transferred upon the Southern face thereof, which may easily bee don by making a Table out of the limb of the Instrument of the several distances of the hour-lines and substiler from the Perpendi­cular line of the Plane. Thus you shall finde

[Page 19]

• The hour of—5:75:00″—11:10:30″
• The hour of—6:57:20″—12 Perpendicular
• The hour of—7:45:00″—1:13:20″
• The hour of—8:34:00″—2:29:45
• The hour of—9:26:00″—3:52:30″
• The hour of—Substiler line—22:20″—4:79:40″
• The hour of—10:18:00″

Now if you would know how to set off the hour-lines which are in use for the North-westerly face, which is the back or contrarie face to the proposed upon that face of your Plane, which hour lines really by the Instrument you have not.

Sweep a circle as I have don in the lesser Diagram of this n^o A, viz. A B A E, and set therein by help of the Table above, all the hour-lines from 5 in morn, to 4 afternoon, both included; in the lower semicircle A E A just as they lie in the Instrument, which will bee the perfect Diall for the South-easterlie face proposed.

Now if you continue these 12 hour-lines through the center to their opposite quadrants in the upper semicircle A B A, you shall di­vide the whole circle into 24 hours, that so you may take thereout the hour-lines which are in use for the Northwesterlie face. As I have don and distinguished by the two Scroles; whereof the greater or outermost Scrole is the Diall for the face proposed, and in the lesser Scrole are the hour-lines for the contrarie face of the same Plane.

And when you pleas to transfer these hour lines in the lesser scrole which are for the contrarie face, upon that face of your Plane, you must place them contrarie to what they are in the lesser scrole, becaus they are the hour-lines for the contrarie face to the proposed, as I have don in the little rundle n^o 2, beeing the perfect Diall for the Northwesterlie, or contrarie face, as the little rundle n^o 1 is the Diall for the proposed.

The better to conceiv a reason hereof, imagin the two little rundles n^o 1 n^o 2 were both cut out of the Paper, and their backsides clapt together, their Horizontal and Perpendicular lines agreeing aright, and center opposite to center, then you will plainly perceive, that the [Page 20] Substiler line K z m in the rundle number 2 will bee one line with the Substiler line L z n in the rundle n^o 1.

As also the hour-lines will meet, and make one line with each hour-line of its opposite quadrant, and this is general for all Planes whatsoever.

An explanation of the Diagrams N^o B & B.

THe second example of an Upright Declining Plane proposed to decline 63, 0″0 Westerly from South.

Here you have

• A z A, the Horizontal lines of this Plane.
• B z B the perpendiculer lines of this Plane.
• C, the place of the Pole, becaus the Southwesterly face is proposed, beeing in the Southwest quarter.
• C L P E, the proper Meridian of the Plane.
• L, the point of this Meridians crossing, or cutting the Plane.
• C, L, K, the subtiler line.
• A z A, servs for the occult line, beeing 90, 0″0, from C, or E.
• z L, 36, 0″0, the Meridian distance from z in the occult line.
• z h, 54, 00 the Pole of this proper Meridian at h.
• n, o, in the limbe, beeing the measure of the arch of this proper Me­ridian, betwixt P & L, is 16. 30″,
• The Poles height above this Plane 16. 30″.

The hour-lines in use for the Southwesterly face, are from a little past 10, or indeed from 11 to 8 (both included) becaus 11, is the first perfect, or real hour-line. And for the Northeasterly or con­trarie face, all the hour-lines from 4 in the morn, to 11 before noon, both included.

Here you have the hour-line of 9 at night given you by the Instru­ment for one of the 12 which for this Southwesterly face is useless, in respect the Sun continues not so long above our Horizon, yet it must not bee left out of either the Table, or greater rundle of the lesser Diagram n^o B, becaus without it you cannot divide the whole rundle (which represents the Plane) into 24 hours, which beeing don, you will bee enabled thereby to take thereout, such hour-lines [Page 21] as are in use for the Northeasterly or contrarie face; as the lesser scrole in the lesser Diagram n^o B shews, but they are to bee placed contrarie to what they are in the lesser scrole, as the little rundle, for the Northeasterly face shews, by the reason formerly given.

• The hour of—10:83:00″—35:40″ Substiler.
• The hour of—11:33:30″—5:38:00″
• The hour of—Perpendicular 12:00; 00″—6:41:45
• The hour of—1:15:15—7:47:30″
• The hour of—2:23:00″—8:55:20″
• The hour of—3:29:00″—9:69:00″
• The hour of—4:33:30″—

An Explanation of the Diagrams N^o C & C.

THe third example of an Upright Declining Plane proposed to decline 63:00″ Easterlie from North.

Here you have

• A z A the Horizontal lines of this Plane.
• B z B the Perpendicular lines of this Plane.
• C the Pole of the Plane in the North-east quarter of the Instru­ment, because the Northeast face is set for principal.
• C L P E the proper meridian of this Plane.
• L the point of this Meridian's crossing the Plane.
• C L K the Substiler line.
• A z A serv's for the occult line 90:00″ from C or E.
• z L 36:00″ the Meridian's distance from z in the occult line.
• z h: 54, 00″ the Pole of this proper Meridian at h.
• n o in the limb, beeing the measure of the Arch of this proper Meridian, betwixt P and L is 16:30″.
• The Pole's height above this Plane—16:30″.

The hour-lines in use for the North-easterly face, are from 4 in the morn, to 11 before noon (both included); and for the South­westerly, [Page 22] or contrarie face; all the hour-lines from a little after 10, or indeed from 11 to 8 (both included) becaus the hour-line of 11 is the first real or perfect hour-line.

Note that these two Planes n^o B & n^o C are both one and the same Plane, onely here is all the difference, that in the Diagram n^o B the South-westerly face is proposed for principal; and in the Di­agram n^o C the North-easterly face is set for principal, as by the Poles of each face doth appear; by which you may plainly perceiv, that what face soever of any Plane is set for principal, the hour-lines and Substiler thereof, beeing taken out of the Instrument just as they lie therein, shall bee the perfect diall, without any alteration for the face proposed; but the hour-lines for the contrarie face must bee con­trarily plac'd, &c.

All which doth most plainly appear in the two lesser Diagrams, n^o B n^o C.

Furthermore, becaus these two Planes have the same Declination, therefore at one working you have the hour-lines, Substilers and and Poles height for both, as by the Tables and hour-lines in use for each do appear, all agreeing in every respect the one with the other.

• The hour of—10:83:00″—:35:40″ Substiler.
• The hour of—11:33:30″—5:38:00″
• The hour of—Perpendicular 12. 00:00″—6:41:45
• The hour of—1:15:15—7:47:30″
• The hour of—2:23:00″—8:55:20″
• The hour of—3:29:00″—9:69:00″
• The hour of—4:33:30″—

An explanation of the Diagrams N^o D & D.

THe first example of a Declining Inclining Plane, falling be­twixt the Pole and Horizon:

Proposed to Decline 24:20″ Westerlie from the South.

Proposed to Incline 36:00″ Northerlie.

Here you have likewise

A z A the Horizontal line of the Plane.

B z B the line crossing the Horizontal at right angles.

z T 54:00″ from z beeing the complement of Inclination.

A T A the Declining Inclining Plane proposed.

z m 36:00″ from z beeing the place of the Pole of the Plane at m.

q m i L q the proper Meridian of the Plane, beeing in this Diagram the same with the hour-circle of one.

D z D the occult line 90:00″ from either q in the limb.

z i 9:00″ the Meridian's distance from z in the occult line.

z h 81:00″ from z the Place of the Pole of this Meridian at h.

L the point of this Meridian's crossing the Plane.

m L K the Substiler line, and hour-line of one also.

n o in the limb, beeing the measure of the arch of this proper Me­ridian betwixt P and L is 17:00″.

The North-pole's height above the upper or South-westerlie face is 17:00″, so much also must the South-pole bee elevated above the lower or North-easterlie face, as by the situation of the Plane in the Diagram doth appear.

The hour-lines in use for the upper face are from 7 in the morn, to 7 in the evening (both included).

And for the lower face from 4 in the morn to 7 before noon (both included) and onely 8 at evening, when the Sun is in the Summer Tropique.

[Page 24]

• The hour of—8:63:00″—2:10:40″
• The hour of—9:42:20″—3:5:30″
• The hour of—10:32:00″—Perpendicular
• The hour of—11:25:00″—4:1:40″
• The hour of—12:20:00″—5:12:40″
• The hour of—Substiler —1:15:40—6:33:50″
• ———7:75:30″

An explanation of the Diagrams N^o E & E.

THe second Example of a Declining Inclining Plane falling to the Southwards of the Zenith.

Proposed to Decline 45:00″ Westerly from South.

Proposed to Incline 38:30″ Southerlie.

Here you have

A z A the Horizontal line of this Plane.

B z B the line crossing the Horizontal at right angles.

z T 51:30″ from z, beeing the complement of Inclination.

A T A the Declining Inclining Plane proposed.

z m 38:30″ from z, beeing the place of the Pole of the Plane at m.

q m i L q the proper Meridian of this Plane.

D z D the occult line 90:00″ from either q in the limb.

z i 36:00″ the Meridian's distance from z in the occult line.

z h 54:00″ from z the Plane of the Pole of this Meridian at h.

L the point of this Meridian's crossing the Plane.

m L K the Substiler line.

n o in the limb, beeing the measure of the arch of this proper Me­ridian betwixt P and L is 62:40″.

The North-Pole's height above the upper face 62:40″.

Here in this Diagram n^o E, when the Sun is in the Summer Tro­pique, it will begin to appear upon the upper, or Northwesterlie face of this Plane, just at 7 in the morn, but I have made the hour-lines in use for this upper face, from 8 in the morn, to 8 at evening (both included) [Page 25] becaus the hour-line of 8 in the morn is the first reall or perfect hour-line.

And the hour-lines in use for the lower or South-easterlie face are from 4 in the morn, when the Sun is in the Summer Tropique to the point f in the Winter Tropique, which is about half an hour past 2, therefore from 4 to 2, (both included).

Here, as I have formerly said, (although in this Diagram) the North-west face is the face proposed and set down, as the Pole of its Plane at m, in the North-west quarter of the Instrument plainly shew's, yet the hour-lines in and by the Instrument are all drawn for the South-east, or lower face, as the lesser scrole in the lesser Diagram n^o E shew's, beeing the same with the hour­lines and Substiler in the Instrument, viz. from 4 in the morn, to 3 afternoon:Now if by these you divide your plane into 24 hours, all the hour-lines which shall bee in use for the upper face, viz. from 8 in the morn to 8 in the evening (both included) shall bee the per­fect dial for this upper face, becaus it is the face set in the Instru­ment for principal, as the greater scrole and little rundle n^o V shew's, beeing all one.

But the hour-lines which are in use for the lower face, from 4 to 2, must bee contrarily plac'd, &c. as the little rundle n^o L shew's, and this is general for all Southerlie Incliners whatsoever.

• The hour of—4:83:00″—Perpendicular,
• The hour of—Substiler line—72:40″—10:4:30″
• The hour of—5:69:30″—11:21:00
• The hour of—6:56:00″—12:38:00
• The hour of—7:42:00″—1:54:00″
• The hour of—8:27:40″—2:69:30″
• The hour of—9:12:00″—3:83:30″

An explanation of the Diagrams N^o F & F.

THe third Example of a Declining Inclining Plane, falling be­twixt the Pole and Zenith.

Proposed to Decline 24:20″ Westerlie from South.

Proposed to Incline 75:00″ Northerlie.

Here you have

A z A the Horizontal line of the Plane.

B z B the line crossing the Horizontal at right angles.

z T 15:00″ beeing the complement of Inclination.

A T A the Declining Inclining Plane proposed.

z m 75:00″ from z, beeing the place of the Pole of the Plane at m.

q m i L q the proper Meridian of the Plane.

D z D the occult line 90:00″ from either q in the limb.

z i 15:00″ the Meridians's distance from z.

z h 75:00″ from z the place of the Pole of this Meridian at h.

L the point of this Meridian's crossing the Plane.

m L K the Substiler line.

n o in the limb, beeing the measure of the arch of the proper Me­ridian, betwixt P and L is—20 00″.

The North Pole's height above the lower or Northeasterlie face 20:00″; therefore the South Pole must bee elevated so much above the upper or South-westerlie face thereof.

The hour-lines in use for the upper face, are from 8 in the morn to 7 in the evening (both included).

And for the lower face, from 4 in the morn to 8 afore noon (both included) and onely 8 in the evening, when the Sun is in the Sum­mer Tropique.

[Page 27]

• The hour of—8:60:00″—16:20 Substiler.
• The hour of—9:27:00″—2:17:30″
• The hour of—10:10:20″—3:22:30″.
• The hour of—Perpendicular—4:29:40″.
• The hour of—11:00:40″—5:38:20″
• The hour of—12:07:00″—6:52:30″.
• The hour of—1:12:30″—7:79:30″.

An explanation of the Diagrams N^o G & G.

THe fourth example of a Declining Inclining Plane, falling near unto the Winter Tropique.

Proposed to Decline 24:20″ Easterlie from North.

Proposed to Incline 14:20″ Southerlie.

Here you have

A z A the Horizontal line of the Plane.

B z B the line crossing the Horizontal at right angles.

z T 75:40″ from z, beeing the complement of Inclination.

A T A the Declining Inclining Plane proposed:—

z m 14:20 from z, beeing the place of the Pole of the Plane at m.

q m i L q the proper Meridian of the Plane.

D z D the occult line 90:00″ from either q in the limb.

zi 8:00″ beeing the Meridian's distance from z.

z h 82:00″ from z, beeing the place of the Pole of this Meridian at h.

L the point of the Meridian's crossing the Plane.

m L K the Substiler line.

n o in the limb, beeing the measure of the arch of this proper Me­ridian, betwixt P and L is 116:00″ Southerlie, from P to the edg of the Plane at L.

Here you must remember what was said at first, that every Plane cut's or divide's the globe of the Univers into two equal halvs, which considered, if you take 116 the distance of the North-pole [Page 28] from the Southern edg of the Plane at L out of 180:00″ the semi­circle, there will remain 64:00″ for the elevation of the same Pole above the Northern edg of the same face, beeing the upper face of the Plane. North-pole's height above the upper face 64:00″.

Now the situation of this Plane in the instrument you see, that all the while the Sun hath above 15:00″ North Declination, it shine's upon the upper face of this Plane, from its rising to its setting (when the daies are at longest with us) for you see the parallel of 10:00″ North Declination, crosseth the Plane but a very small time be­fore the Sun setting, when it is therein.

So that for the upper or North-easterlie face you must have all the hour-lines from 4 in the morn to 8 at evening (both included).

And for the lower or South-westerlie face, first 8 and 9 in the morn, and then from 2 to 7 in the evening, which hour-lines will bee of very small use, becaus the Plane incline's so much Souther­ly.

Note likewise, that this Plane is a Southerlie Declining Inclining Plane, as the Plane in the Diagram n^o E is, therefore the same me­thod is to bee observed, as is there taught, which is also most plain­ly demonstrated in the lesser Diagram of this number G.

• The hour of—8:79:10″—Perpendicular.
• The hour of—9:64:30″—2:4:40″
• The hour of—10:50:30—3:20:00″
• The hour of—11:37:20″—4:35:40″
• The hour of—Substiler—34:20″—5:52:00″
• The hour of—12:24:00″—6:69:00″
• The hour of—1:9:30″—7:85:10″

An explanation of the Diagrams N^o H & H.

BEeing an example of a Meridian Inclining Plane, proposed to Incline 53:00″ Westerlie.

I have said before, that for all Planes of this nature, the Meridian line, or North and South line in the Instrument N z S must bee one and the same line with, and serv for the Horizontal line of the Plane; And therefore the Prime Vertical, or East and West line in the In­strument W z E stand's for the line which must cross the Horizontal line of the Plane at right angles:The rest of the work differ's no­thing from the former Declining Inclining Planes.

So that here the Meridian

N z S is the Horizontal line of this Plane.

W z E stand's for the line crossing the Horizontal at right angles.

z T 37:00″ from z, beeing the complement of Inclination.

N T S the Meridian Inclining Plane proposed.

z m 53:00″ from z, beeing the place of the Pole of this Plane at m.

q m i L q the proper Meridian of this Plane.

D z D the occult line 90:00″ from either q in the limb.

z i 35:00″ this Meridian's distance from z in the occult line.

z h 55:00″ from z the place of the Pole of this Meridian at h.

L the point of this Meridian's crossing the Plane.

m L K the Substiler line.

n o in the limb, beeing the measure of the arch of this proper Me­ridian betwixt P and L is 28:30″.

The North-pole's height above the upper face 28:30″.

The hour-lines in use for the Eastern or upper face, are from 4 in the morn to 2 afternoon (both included).

And for the Western or lower face, all the hour-lines from 1 to 8 in the evening (both included).

[Page 30]

• The hour of—12:90:00″—6:32:40″
• The hour of—9:65:00″—7:40:00″
• The hour of—2:34:45—:44:30″ Substiler
• The hour of—3:7:50″—8:47:20″
• The hour of—Perpendicular.—00:00″—9:54:20″
• The hour of—4:10:45—10:63:00″
• The hour of—5:23:20″—11:74:15
• If the Plane shall incline Easterlie as this doth Westerlie, the same order in working is to bee observed in every respect.

An Explanation of the Diagrams N^o I & I.

THe first example of a Polar Declining Plane,

Proposed to Decline 52:00″ Westerlie from South.

Proposed to Incline 64:00″ Northerlie.

In these Polar declining Planes, you are to finde out by the In­strument two things in chief, and then the rest of the work will dif­fer nothing from the making of an ordinarie Meridian, or direct Polar Diall.

The first is the Arch betwixt the Substiler and the Horizontal line of the Plane in degrees and minutes.

The second is the Inclination or distance betwixt the proper Me­ridian of the Plane, and the Meridian of the place; both which will bee easily given you by the Instrument.

Thus I set down upon the Instrument according to the Declinati­on and Inclination proposed.

• A z A the Horizontal line of the Plane.
• B z B the line crossing the Horizontal at right angles.
• z T 26:00″ from z, beeing the complement of Inclination.
• A P T A is the Plane proposed, which in this Diagram fall's to bee [Page 31] the same with the Meridian or hour-cirlce of 9, cutting or crossing the Pole of the Instrument P; and therefore must bee a Polar Plane, and neither Pole elevated above either face.
• z m 64:00″ from z, beeing the place of the Pole of this Plane at m, which fall's just upon the hour-circle of three in the afternoon, and therefore the Substiler, and hour-line of 3 must bee one and the same line, and serv for both; as formerly in the Diagram n^o D was de­monstrated.
• q P m q the proper Meridian of this Plane.

Now (as in every of the former Diagrams) if from the Pole of the Plane m, through the point of this Meridian's crossing the Plane, which is at P the Pole of the Instrument, you draw a right line to the limb, it must bee the Substiler line, and hour line of three also, which is represented by the line m P K, whose Horizontal Arch be­twixt A and K in the limb, is 61:00″ which is the first thing requi­red.

The next is to observ where the Pole of your Plane fall's to bee either upon an hour-circle (as here it doth) or betwixt any two, for the distance thereof from the Meridian of the Plane (which in the Instrument is ♣ z S) beeing taken and converted into degrees and minutes, shall give the Inclination or distance betwixt the pro­per Meridian of the Plane, and the Meridian of the place, which is here represented by the Arch of the Equator ♈ m in the South-west quarter of the Instrument, and therefore 3 hours of time, or 45:00″ from the North and South line in the Instrument, which is the Me­ridian of the place; And this is the second thing required.

A K 61:The Horizontal Arch.

♈ m 45:00″ Inclination or distance betwixt the two Meridians.

Here the South-west face of this Plane is proposed for principal or upperface.

The hour-lines in use for which face, are from 10 before noon, to 8 in the evening (both included).

And for the lower face from 4 in the morn, to 8 aforenoon (both included).

Now to take the Table out of the Instrument, observ the place of the Pole of your Plane (which in these planes will alwaies fall up­on the Equator) observ I saie, the place of the Pole of your Plane, [Page 32] amongst the Meridians or hourcircles, which justly reckoned either waie to the next hour-circles one by one, by allowing 14:00″ of the Equator for every hour of time, (and so proportionably for any part of an hour, if the Pole fall betwixt two meridian or hour­circles) you may make a Table thereby, to transfer your hour-lines upon your Plane, which must bee the several distances of the hour­lines from the Substiler line of the Plane.

• The Substiler & hour of—3:00″:00″
• The Substiler & hour of—4 & 2:15:00″
• The Substiler & hour of—5 & 1:30:00″
• The Substiler & hour of—6 & 12:45:00″
• The Substiler & hour of—7 & 11:60:00″
• The Substiler & hour of—8 & 10:75:00″

Now for delineating your diall upon your Plane, (the Table bee­ing made) first, as in the lesser Diagram of this number I, I draw the Horizontal line A A, next at anie convenient point therein, viz. at B; I sweep the Arch C D. and finding out a chord of 61:00″ for the arch betwixt the Horizontal line and Substiler, I put it over from C to E, through which point E, I draw the Substiler line BHEF, which must be the hour-line of 3 also, to which all the rest of the hour-lines must be parallel. Next at som convenient point in the Substiler line, viz. at H, I sweep a semicircle, whose semidiameter H I, I proportion to bee the height of the Stile above the Plane; then through this point I, at right angles to the Substiler line, I draw the line G I G, & another line parallel to this A H L, by which two parallel lines, you shall set down your hour-lines upon your Plane, by the Table and this Canon.

The Canon.

As H I Radius to I K the Tangent of 75:00″, beeing the distance of the hour-lines of 8 & 10 from the Substiler.

So is H I 50/100 of an inch to I K 1 87/100 of the same measure to bee set off from eitherside the substiler, in the two parallel lines and so draw the lines through the Plane, as the line 8 K 8 shew's. And so for all the rest the same method hold's.

An explanation of the Diagrams K & K.

THe second Example of a Polar Declining Plane,

Proposed to Decline 62:30″ Easterly from South.

Proposed to Incline 70:00″ Northerly.

According to which proposal I draw

A z A for the Horizontal line of the Plane.

B z B for the line crossing the Horizontal at right angle.

z T 20:00″ from z, beeing the complement of Inclination.

A T P A is the Polar Declining Plane proposed, becaus it passeth through the Pole of the Instrument P.

z m 70:00″ from z, the place of the Pole of this Plane at m.

q P m q the proper Meridian of this Plane.

P the point of this Meridian's crossing the Plane.

m P K the Substiler line, whose Horizontal Arch from A to K is 56:00″ in the limb of the Instrument.

The Inclination or distance betwixt the proper Meridian of the plane, and the Meridian of the place is the Arch of the Equator v m, beeing in time about three hours 48:00″ before noon, which is 12:00″ after 8 in the morn, or 57:00″ in measure, as the Pole of the plane at m shew's.

The Horizontal Arch 56:00″.

Distance betwixt the two Meridians 57:00″.

The hour-lines in use for the upper or Southeasterlie face are, from 4 in the morn, to one after noon (and no more) except the plane bee very large. And for the lower face from 3 to 8 at evening (both included).

[Page 34]

• The hour of—3:78:00″—Substiler line.
• The hour of—4:63:00″—9:12:00″
• The hour of—5:48:00″—10:27:00″
• The hour of—6:33:00″—11:42:00″
• The hour of—7:18:00″—12:57:00″
• The hour of—8:03:00″—1:72:00″
• —2:87:00″

Thus the Table beeing made, the delineation or drawing of your lines upon the Plane, differ nothing from that of the former Dia­gram.