Fig. 21 de. 30 m. shall serve for a South Dial Reclining 60 deg. in LIX. the Latitude of London.

III. Variety. If a South Plain in the Latitude of London should Recline from the Zenith thereof 38 d. 30 m. Equal to the Complement of the Latitude of London. Then

The Difference between the Complement of the Latitude of London and the Reclination being nothing; it thews, the New-Latitude to beno Latitude, that is neither Pole hath any Elevation above such a Plain: And therefore a Dial for fuch a Plain must be made in all refpects as an Erect Direct East or West Dial is made, by the Precepts in Chapter VII. hereof: Only, the Hour-Line of VI there, must be the Hour-Line of XII in this: And as the Stile there was equal to the distance between Six and Three or Nine a Clock. So in this, it must be equal to the distance between XII and IX or III. And may be either a Plate of that breadth; or a Wyre or Pin of that Length.

II. Of North Recliners

I. Variety. If a North Plain in the Latitude of 51 de. 30 m. should Recline from the Zenith 20 deg. In what Latitude will that be an Horizontal Plain?

The Reclination 20 deg. being less than the Complement of the Latitude of London, 38 de. 30 m. Add the Reclination 20 de. and the CoLatitude 38 deg. 30 m. together; their Sum 58 deg 30 m. is The New Latitude: And an Horizontal Dial for that Latitude shall be a North Plain Reclining 20 deg. in the Latitude of 51 de. 30 m.

II. Variety. If a North-Plain in the Latitude of London, 51 deg. 30 m. should Recline from the Zenith 75 deg. In what Latitude will fuch a Plain be Horizontal?

The Reclination 75 deg. being greater than 38 deg. 30 m. Add them together, and they make 113 deg. 30 min. which being above 90 deg. take the Complement thereof to 180 deg, which is 66 deg. 30 m. And that is the New-Latitude: So that an Horizontal Dial made for the Latitude of 66 de. 30 m. will bea North Plain Reclining 75 deg. in the Latitude of London 51 deg. 30 m.

III. Variety. If a North Plain in the Latitude of London 51 de. 30 m. should Recline from the Zenith thereof 51 de.

30

30 m. In what Latitude will sucha Plain be Horizon- Fig.

tal?

Here, the Reclination 51 de. 30 m. is Equal to the Latitude of London, And the Sum of the Reclination 51 de. 30 m. and the Complement of the Latitude 38 de. 30 m. Added together, their Sum is 90 deg. for The New-Latitude: And an Horizontal Dial made for that Latitude of 90 deg. (which is no other then a Cirole divided into 24 Equal Parts, for the Hours; and a Wyre erect-ed perpendicularly, of any Length for the Stile.)

These are all the Varieties of South and North Reclining Plains, Reduced to New Latitudes, wherein they will become Horizontal Plains: So that neither Trigonometrical Calculation, Geometrical Projection, or Operation by the Globe, need be here Repeated, they being all the fame as in the IVth Chap-ter hereof..

Note also: That in making of any of these North or South Reclining Dials, you have made also A Direct North or South Dial inclining from the Zenith towards the Horizon

Re

In

fo many Degrees as as is the clination. So that when
you have made a South-Dial Reclining from the Zenith 60
deg. (as is the Second Variety of South Recliners in this Chap-
ter) you have made also a North Dial inclining to the Ho--
rizon 60 degrees, either by drawing of the Hour-lines
and Stile through the Centre; or by turning the Reclining
Dial about upon the Hour-line of VI. And then, as the
North Pole is elevated upon the South Recliner; fo much
will the South Pole be elevated above the North Incli--
ner, &c.

LIX..

CHAP. ΧΙ.

Of Direct East or West Reclining Dials, and how Hour-lines may be defcribed upon them.

A

duced

S all Direct North and South Reclining Dial Plains were Re-to New-Latitudes wherein they would be Horizontal Plains; and therefore made by the Directions given in the CHAP. IV. hereof: So all Direct East or West Reclining Dial

Plains

1

290 Fig. Plains in any one Latitude, may be Reduced to Erect, or Upright, LIX. Declining Plains in another Latitude: and therefore may be made by the Precepts delivered in the VIIIth. CHAP. hereof. Either, By the Globe; By Spherical Projection; or, By Trigonometrical Calculation: So that the Work of this CHAP. shall be only to shew

I. How to Reduce any Direct East or West Reclining Dial Plain
in any Latitude, (suppose London 51 de. 30 m.) to a
New-Latitude: wherein the Reclining Plain shall become
an Erelt (or Upright) Plain. And

If What Declination, that Upright Plain shall have in that
New-Latitude.

EXAMPLE.

Suppose then, that a Direct East or West Plain, in the Latitude
of 51 de. 30 m. should Recline from the Zenith 40 deg.
In what Latitude will that be an Upright Plain? And what
Declination shall it have in that Latitude?

RULE.

The Complement of the known Latitude, is (always) The New-
Latitude. And

The Complement of the Reclination is (always) The Declination
in that New-Latitude.

So that if an East or West Dial, should Recline 40 deg. in the Latitude of 51 d. 30 m. That will be An Upright Plain, Declining 50 deg. in the Latitude of 38 de. 30 m

For, 38 de. 30 m. being the Complement of the Latitude of London 51 d. 30 m. is the New-Latitude: - And 50 deg. being the Complement of 40 de. the Plains Reclination, is the Declination in the New-Latitude.

So that, if (according to the Precepts delivered in the VIIIth. CHAP. hereof) you make an Upright Dial for the Latitude of 38 de. 30 min. to Decline 50 deg. Such a Dial will ferve for an Eaft or West-Dial Reclining 40 de. in the Latitude of (London) 51 de. 30 m.

Thus for the Making of the Dial.

But in the placing of the Dial (thus made) upon the Reclining Plain, this Difference is to be observed: For, Whereas in all Upright Declining Plains, the Meridian, or Hour-line of XII is always Perpendicular to the Horizon of the Place for which it is made.

But

291 But this Declining Plain when it is applyed to the Reclining Plain, Fig. the Hour-line of XII. must lie Parallel to the Horizon of the Place, LX. as in the Figure.

"

And here Note, that all East Recliners in the known Lati-
tude (as here London)are North-East Decliners in the New
Latitude: And all West-Recliners, are North-West-Decli-

ners.

;

And Note farther; That upon all East and West Reclining
Plains in North-Latitudes, that the North Pole is (always)
Elevated: And upon the East and West inclining Plains,
opposite' to them, the South Pole is Elevated.

And Lastly, Note that when you have made an East or West
Reclining Dial, you have made Four, viz. An East and
West Reclining, and, An East and West Inclining.

CHAP. XII.

Of Dial Plains that do both Decline and Recline: How
Hour-Lines, &c. are to be described on them.

T

HESE Plains may also be Reduced to New-Latitudes and New Declinations, where they may stand as Upright Decli ners: And so Dials may be described on them, by the Directions in the VIIIth. CHAP. hereof: And to find the New-Latitude, and New Declination of any such Plain, The Rules following will

direct.

EXAMPLE.

Suppose that in the Latitude of London, 51 d. 30 min: A South Plain should Decline towards the East or West 24 deg. 20 min. And also Recline from the Zenith 54 deg.

I. To find the New-Latitude

The Canon for Calculation.

As the Radius, Sine 90 de.

So is the Co-Tangent of the Reclination 54 d.

To a Fourth Tangent, viz. 33 de. 30 m.

Is to the Co-fine of the Old Decl. 24 de. 20 m.

10.

9.959596

9.861261

49.820857

This Fourth Tangent being thus found, the Rules following

are to be observed.

I. In South Recliners.

RULE I. This Fourth Tangent must be compared with the Old Latitude, and the Complement of their Difference is the NEW LATITUDE.

i

So, In this Example, The Fourth Arch 33 de. 30 m. being fubftracted from the Old Latitude 51 de. 30 m. their difference is 18 deg. whose Complement 72 deg. is the NEW LA TITUDE.

RULE II. If the Fourth Tangent fall out to be Equal to the Old Latitude, Then the Difference will be nothing; And so the Plain will be a Polar declining Plain: For the Pole will have no Elevation over it.

So, in the Latitude of 51 de. 30 m. If a South Plain should Decline towards the East or West 65 deg. 40 m. And Recline 18 de. 9 m. By the former Canon, the Fourth Tangent will be found 51 de. 30 m. Equal to the Old Latitude: So that the Difference is. Nothing. And the Hour-Lines will be Parallel one to another as in the Direct Eaft, West, and Polar Dials, in CHAP. VII.

and CHAP. XV

RULE IH. If the Fourth Tangent prove to be Greater than the Old Latitude, Then, The North Pole is Elevated in) South Decliners: But if the Fourth Tangent be Leffer than the Old Latitude, Then, The South Pole is Elevated in North Decliners.

II. In North Recliners.

RULE I. The Fourth Tangent found as before, is to be compared with the Complement of the Old Latitude, and their Difference is the NEW LATITUDE. RULE II. If the Fourth Tangent prove to be Equal to the Complement of the Old Latitude: that Declining Reclining Plain will be an Æquinoctial Plain Declining.

So, In the Latitude of 51 de. 30 m. If a Plain should Decline from the North towards the East or West 60 deg. And alfo Re-cline from the Zenith 32 deg. 11 m. The Fourth Tangent will be found to be 38 de. 30 m. Equal to the Complement of the Old Latitude: And will, therefore, be An Equinoctial Declining Plain..

:

ΙΙ. Τρ