The sector on a quadrant, or A treatise containing the description and use of four several quadrants two small ones and two great ones, each rendred many wayes, both general and particular. Each of them accomodated for dyalling; for the resolving of all proportions instrumentally; and for the ready finding the hour and azimuth universally in the equal limbe. Of great use to seamen and practitioners in the mathematicks. Written by John Collins accountant philomath. Also An appendix touching reflected dyalling from a glass placed at any reclination.

Of the Stars graduated on the PROJECTION.

SUch Stars as are between the two Tropicks only, are there in∣scribed, and such haue many things common in their Motion with the Sun when he hath the like Declination, as the same Am∣plitude, Semidiurnal, Arke, Meridian, Altitude, Ascentional diffe∣rence, &c.

These Stars have Letters set to them to direct to the Circle of As∣censions on the back of the Quadrant, where the quantity of their right Ascension, is expressed from one of the Equinoctial points; those that have more Ascension then 12 hours from the point of Aries, are known by the Character plus + set to them; many more Stars might be there inserted, but if they have more then 23^d 31′ of Declination, the Propositions to be wrought concerning them are to be performed with Compasses, by the general Lines on the Quadrant.

To find the true time of the Day or Night when any Star commeth to the Meridian.

In the performing of this Proposition we must make use of the Suns whole right Ascension in time, which how that might be known hath been already treated of, as also of the Stars whole right As∣cension, which may be had from the Circle of Ascensions on the back of the Quadrant if 12 hours be added to the right Ascension of a Star taken thence that hath a Letter Character† affixed to it.

Substract the Suns whole right Ascension from the Stars whole right Ascension, encreased by 24 hours when Substraction cannot be made without it, the remainder is less then 12 shews the time of the afternoon or night when the Star will be upon the Meridian; but if there remain more then 12, reject 12 out of it and the residue shews the time of the next morning when that Star will be upon the Meridian.

Example.

The 23^d of December the Suns whole right Ascension is 18 hours 53′ which substracted from 4 ho: 16′ the right Ascension of the Bulls eye encreased by 24 there remains 9^h 23′ for the time of that Stars comming to the Meridian, and being substracted from 6 ho: 30′ the right Ascension of the great Dogg, there rests 11 ho: 37′ for the time of that Stars coming to the Meridian at night.

This Proposition is of good use to Sea-men, who have occasion to observe the Latitude by the Meridian Altitude of a Star, that they may know when will be a fit time for observation.

In finding the time of the Night by the Stars, we use but 12 ho: of right Ascension, nor no more in finding the time of their rising or setting, so that when it is found whether it be morning or even∣ing is left to judgement, and may be known by comparing it with the former Proposition, if there be need so to do.

To find the Declination of any of these Stars.

This is engraven or annexed to the Stars names, yet it may be found on the Projection, by rectifying a Bead to the proposed Star, and bringing the Thread and Bead to that Ecliptick it wil intersect; and in the same Position the Thread will intersect the said Stars declination in the Quadrant of Declinations; if the Bead meet with the Sum∣mer Ecliptick the Declination is North, if with the Winter South.

To find the Amplitude and Ascensional difference of any of the Stars on the Projection.

BRing the Bead rectified to the Star to either of the Horizons, the Thread being kept in its due Extent, and where it in∣tersects the same it shews that Stars Amplitude which varies not, and is Northward if the Star have North declination, otherwise South∣wards, the Thread likewise intersecting the Limb, sheweth the Stars Ascensional difference.

Example.

So the Bead being rectified to the Bulls eye, and brought to the lower Horizon, shews the Amplitude of that Star to be 25^d 54′ Northwards because the Star hath North Declination; And the Thread lyeth over 20′ 49′ of the Limb which is this Stars Ascensio∣nal difference, which in Time is 1 ho 23^m The Thread in the Limb lyeth over 4 ho 37^m from midnight for the Stars hour of rising, and over 7 ho 23^m from the Meridian for the Stars hour of setting always in this Latitude which with the Amplitude varies not, except with a very small allowance in many years.

To find a Stars Diurnal Ark, or the Time of its continu∣ance above the Horizon.

When the Star hath

• North Declination add the Ascensional difference of the Star before found in Time to 6 hours, the Sum is half the time.
• South Declination Subtract. the Ascensional difference of the Star before found in Time from 6 hours, the Residue is half the time.

Of that Stars continuance above the Horizon, which doubled, shews the whole time, the Complement wherof to 24 ho is the time of that Stars durance under the Horizon.

Example.

So the Ascensional difference of the Bulls eye being in time 1 ho: 23 added to 6 hours, and the Sum doubled makes 14 hours 46^m for the Stars Diurnal Ark or abode above the Horizon, the residue whereof from 24 is 9 ho: 14^m for the time of its durance under the Horizon.

To find the true time of the Day or night, when the Star riseth or setteth.

THe Stars hour of rising or setting found as before, being no other but the Ascensional difference of the Star added to, or substract∣ed

from 6 hours; which the Thread sheweth in the Limb the Bead being rectified to a Star, and brought to that Horizon it will inter∣sect; is not the true time of the night; but by help thereof that may be come by; this we have denominated to be the Stars hour, and is no other but the Stars horary distance from the Meridian it was last upon;

If a Star have

• North Declination the Stars hour of rising must be reckon∣ed to be before 6 and the time of its setting after 6
• South Declination the Stars hour of rising must be reckon∣ed to be after 6 and the time of its setting before 6

Now the time of the Stars rising or setting found by this and the former Propositions must be turned into common time by this Rule. To the Complement of the Suns Ascension add the Stars Ascension, and the Stars hour from the Meridian it was last upon, the Amount if less then 12 shews the the time of Stars rising or setting accordingly; but if it be more then 12 reject 12 as oft as may be, and the remaind-sheweth it.

So upon the 23^d of December for the time of the Bulls eye rising.

	h	m
The Complement of the Suns Ascension found by the foreside of the Quadrant is —	5	7
And the said Stars Ascension on the backside is—	4	16
The Stars hour of rising is—	4	37
14 hours.

From which 12 rejected rests 2 hours for the time of that Stars ri∣sing, which I conclude to be at 2 in the afternoon, because that Star was found to come to the Meridian at 23^m past 9 at night, the like Operation must be used to get the time of that Stars setting, which will be found to be at 4 ho 46^m past in the morning.

	h	m
Complement ☉ Ascension—	5	7
Stars Ascension—	4	16
Stars hour of setting—	7	23
16 h.	46′

To find what Altitude and Azimuth a Star that hath North Declination shall have when it is 6 hours of Time from the Meridian.

REctifie the Bead to the Star, and bring the Bead and Thread to the left edge of the Quadrant, and there among the Parra∣lels of Altitude and Azimuths it sheweth what Altitude and Azimuth the Star shall have.

Example.

So the Bead being set to the Bulls eye, and brought to the left edge of the Quadrant it will be found to have 12′ 17′ Altitude, and 80^d 3′ Azimuth from the South, when it is 6 hours of time from the Meridi∣an, which Proposition is afterwards used to know to which Eclip∣tick in some Cases to rectifie the Bead as hath likewise been intima∣ted before.

The Azimuth of a Star proposed, To find what time of the Night the Star shall be upon that Azimuth, and what Altitude it shall then have.

SUpposing the Azimuth proposed to be nearer the South Meridi∣an then that Azimuth the Star shal have when it is 6 hours from the Meridian: Bring the Bead rectified to the Star, to the proposed Azimuth, and among the Parralels of Altitude it shews that Stars Altitude, and the Thread in the Limb shews that Stars hour to be turned into common time to attain the true time sought.

Example.

If the question were What Altitude the Bulls eye shall have when his Azimuth is 62^d 48′ from South, this being less Azimuth then he hath at 6 hours from the Meridian, the rectified Bead being brought to the Azimuth sheweth among the Parralels the Altitude to be 39^d and the Stars hour shewn by the Thread in the Limb is either 8 ho: 56′ or 3 ho: 4′ from the Meridian; then if upon the 23 of Decem∣ber

you would know at what time the Star shall have this Altitude on this Azimuth, Change the Stars hour into common time by the former Rule.

Decemb. 23	Complement of ☉ Ascension	5h	7′	5h	7′
	Stars Ascension—	4	16	4	16
	Stars hour—	8	56	3	4
		18	19	12	27

And you will find it to be at 19′ past 6 in the evening, or at 27^m past midnight.

For Stars of South Declination being they have no Altitude a∣bove the Horizon at 6 ho: distance from the Meridian, the opera∣tion will be the same, void of Caution.

But for Stars of North Declination when the proposed Azimuth is more remote from the South Meridian then the Azimuth of that Star 6 ho from the Meridian, another Bead must be rectified to the Winter Ecliptick, and carried to the Azimuth proposed above the upper Horizon, where amongst the Parralels it shews the Altitude sought; and the Thread in the Limb sheweth the Stars hour to be converted into common time.

Example.

The Azimuth of the Bulls eye being 107^d 53′ from South, which is more then the Azimuth of 6 hours, the other Bead set to the Winter Ecliptick, and carried to that Azimuth in the Tail, shews the Altitude to be 6^d and the Stars hour to be 5 ho: 18′ Or 6 ho: 42′ which converted into common time, as upon the 23^d of De∣cember, will be either 41^m past 2 in the afternoon, or 5^m past 4 in the morning following.

h	'	h	'
December 23	Complement Suns Ascension	5	7	5	7
	Stars Ascension—	4	16	4	16
	Stars hour—	5	18	6	42
Rejecting 12 the Total is—	2	41 Or 4	5

The Hour of the night proposed to find what Altitude and Azi∣muth any of the Stars on the Projection that are above the Horizon shall have at that time.

FIrst turn common time into the Stars hour, the Rule to do it is, To the Complement of the Stars Ascension add the Suns As∣cension, and the time of the night proposed, the Aggregate if less then 12 is the Stars hour; if more reject 12 as oft as may be, and the remainder is the Stars hour sought. So the 23 of December, at 8 a Clock 59 minutes past at night what shall be the Horarie di∣stance of the great Dogg from the Meridian

Complement of great Doggs	h	'
Ascension—	5	30
Suns Ascension —	6	53
Time of the night —	8	59
The Sum is, 12 rejected —	9	22

Then for Stars of South Declination, rectifie the Bead to the Star proposed, and lay the Thread over the Stars hour in the Limb, and the Bead amongst the Parralels and Azimuths, shews the Alti∣tude and Azimuth of the Star sought. Example. So the Bead be∣ing rectified to the great Dogg, and the Thread laid over 9 ho 22′ in the Limb, the Bead will shew the Altitude of that Star at that time of the night to be 14^d and its Azimuth 39^d from the South.

The Operation is the same for Stars of North Declination when the Stars hour found as before is not more remote from the South Meridian then 6 hours on either side.

But if it be more then 6 ho distance from the Meridian as before 6 after its rising, or after it before its setting, then as before sug∣gested, one Bead must be rectified to the Star, and brought to the Summer Ecliptick, where the Thread being duly extended, another must be set to the Winter Ecliptick, and afterwards the Thread laid over the Stars hour in the Limb, this latter Bead will shew the Stars Azimuth and Parralel of Altitude in the reverted Tail above the up∣per Horizon.

Example.

So upon the 23 of December, I would know what Azimuth and Altitude the Bulls eye shall have at 4 a Clock 5 minutes past the morning following.

Time proposed	4h	5′
Complement of Bulls eye Ascension —	7	44
Suns Asce••sion 23 of De∣cember —	6	53
12 rejected rests —	6h	42′

Proceed then and lay the Thread over 42′ past 6 and the Bead a∣mong the other Paralels in the Tail sheweth the Stars Altitude to be 6^d and its Azimuth from the Meridian 107^d 53′ These two Propositions have a good tendency in them to discover such Stars as are upon the Projection if you know them not, but supposing them known the Proposition of chiefest use is

By having the Altitude of a Star given to find out the true Time of the night, and the Azimuth of that Star.

If the Stars observed Altitude be less then its Altitude at 6 ho: distance from the Meridian; Bring the Bead, rectified to the Star, to the Summer Ecliptick and set another Bead to the Winter E∣cliptick, Then carry it to the Parralell of Altitude above the upper Horizon in the Reverted taile and there it will shew the Azimuth of that Star; and the thread in the Limbe shews the houre.

Example.

So if the observed Altitude of the Bulls eye were 6^d its Azimuth would be found to be 107^d 53′ from the South, and its hour 42′ past 6 from the Meridian the true time would be found to be 5 minutes past 4 in the morning the 24 of December.

	ho:
Complement of ☉ Ascension the 23 of December —	5	7
Stars hour—	6	42
Stars Ascension —	4	16
	4	5

But for Stars that have South declination or north, when their Altitude is more then their Altitude being 6 hours from the Meri∣dian, this trouble of rectifying two Beads is shunned; in this Case only bring the Bead that is rectified to the Star to the Parralel of Altitude, and there among the Azimuths it will shew the Stars Azi∣muth, and the Thread in the Limb intersects the Stars hour sought.

Example.

December 11^th Bulls eye Altitude 39^• Azimuth from the South 62^d 48

Hours from the Meridian—	8h	56′
Complement of ☉ Ascension—	6	00
Ascension of Bulls eye—	4	16
The true time of the night was 12′ past 7 of the Clock	7	12

Another Example.

The great Doggs observed Altitude being 14^d his Azimuth from the South would be 39^d.

h	m
And the Stars hour from the Meridian—	9	22
Stars Ascension—	6	30
If this Observation were upon the 31 of December, the Complement of the Suns Ascension would be—	4	30
	8	22

And the true time of the night 22 minutes past eight of the Clock.

For varieties sake there is also added to the Book a Draught of the Projection for the Latitude of the Barbados; in the use where∣of the Reader may observe that every day when the Sun comes to the Meridian between the Zenith and the Elevated Pole, he will upon divers Azimuths in the forenoon (as also in the afternoon) have two several Altitudes, and so be twice before noon, and twice

[illustration]

[illustration]

afternoon, at several times of the day, upon one and the same Azi∣muth, viz. only upon such as lye between the Suns Coast of rising and setting, and his remotest Azimuth from the Meridian, which causeth the going forward and backward of the shaddow; but of this more hereafter, when I come to treat of Calculating the Suns Altitude on all Azimuths; It may also be observed that the Sun for the most part in those Latitudes hath no Vertical Altitude or De∣pression, and so comes not to the East or West.

Moreover there is added a Draught of this Projection for the Latitude of Greenland, in the use whereof it may be observed that the Sun, a good part of the Summer half year comes not to the Horizon, and so neither riseth nor sets.

And that no convenient Way that this Projection can be made should be omitted, there is also one drawn in a Semi-Circle for our own Latitude, which in the use will be more facile then a Qua∣drant, there being no trouble before or after six in the Summer time, with rectifying another Bead to perform the Ope∣ration in the reverted Taile, neither doth the Drawing hereof oc∣cupy near the Breadth, as in a Quadrant, and so besides the ease in the use is more exact in the performance; there being no other Rule required for rectifying the Bead, but to lay the Thread over the day of the Month, and to set the Bead to that Ecliptick the Thread intersects.

A Semi-Circle is an Instrument commonly used in Surveigh, and then it requires a large Center-hole; however this Projection may be drawn on a Semi-Circle for Surveigh, but when used at home there must a moveable round Bit of brass be contrived to stop up that great Center-hole, in which must be a small Center-hole for a Thread and Plummet to be fastned, as for a Quadrant and some have been so fitted.

The Reader will meet with variety of Lines and furniture in this Book to be put in the Limb, or on other parts of the Semicircle, as he best liketh. The Projection for the Barbados & Greenland, are drawn by the same Rules delivered in the Description of the Quadrant, and so also is the Summer part of this Semi-Circle, and the Winter part by the same Rules that were given for drawing the Reverted Taile.

Of the Quadrant of Ascensions.

The turning of the Stars hour into the Suns hour and the the con∣verse may be also done by Compasses upon the Quadrant of As∣censions on the back side.

To turn the Stars Hour into common time, called the Suns hour.

THe Arithmetical Rule formerly given is nothing but an abridg∣ment of the Rule delivered by Mr. Gunter, and others, and the work to be done by Compasses, differeth somewhat from it, though it produce the same Conclusion which is:

To get the difference between the Ascension of the Sun and the Star by substracting the less from the greater; this remainder is to be added to the Stars hour, when the Star is before, or hath more Ascension then the Sun, but otherwise to be substracted from it, and the Sum or remainder is the true time sought.

To do this with Compasses, take the distance between the Star and the Suns Ascension, and set the Suns foot to the observed hour of the Star from the Meridian it was last upon, letting the other foot fall the same way it stood before, and it sheweth the time sought, if it doth not fall off the Quadrant.

If it doth, the work will be to finde how much it doth excur, and this may be done by bringing it to the end beyond which it falleth, letting the other foot fall inward, the distance then between the place where it now falleth, and where it stood before, which was at the Stars hour, is equal to the said excursion, which being taken, and measured on the other end of the Scale, shews the time sought.

This trouble may be prevented in all Cases, by having 12 hours more repeated after the first 12, or 6 hours more may serve turn if the whole 18 hours be also double numbred, and Stars names be∣ing set to the Additional hours, possibly the Suns Ascension and Star do not both fall in the same 12 hours, yet notwithstanding the distance is to be taken in the same 12 hours between the quanti∣ty of the Suns Ascension and the Stars, and to proceed therewith as before, and the Compasses will never excur; in the numbring of these hours, after 12 are numbred they are to begin again, and are numbred as before, and not with 13, 14, &c.

And this trouble may be shunned when there is but 12 hours by

assuming any hour to be the Stars hour, with such condition that the other foot may fall upon the Line; and the said assumed hour repre∣senting the Stars hour; count from it the time duly in order, till you fall upon the other foot of the Compasses, and you will obtain the true time sought.

To turn common time, or the Suns hour into the Stars hour.

THis is the Converse of the former; take the distance between the Star and the quantity of the Suns Ascension, and set the Star foot to the Suns hour, letting the other fall the same way it stood before, and it shews the time sought.

Of the Quadrat and Shaddows.

Both these as was shewed in the Description of the Quadrant, are no other then a Table of natural Tangents to the Arks of the Limb and may supply the use of such a Cannon, though not with so much exactness, all the part of the Quadrate are to be estimated less then the Radius, till you come against 45^• of the Limb, where is set the figure of 1, and afterwards amongst the shadows is to be account∣ed more then the Radius, and so where the Tangent is in length

• 2 Radii as against 63^d 26^m of the Limb.
• 3 Radii as against 71 34 of the Limb.
• 4 Radii as against 75 58 of the Limb.
• 5 Radii as against 78 42 of the Limb.

are set the figures 2, 3, 4, 5, and because they are of good use to be repeated on the other side of the Radius in the Quadrat, there they are not figured, but have only full points set to them, falling against the like Arks of the Limb from the right edge towards the left, as they did in the shadows from the left edge towards the right.

To find a hight at one Observation.

[illustration]

LEt A B represent a Tower, whose Altitude you would take, go so far back from it that looking through the sights of the Quadrant, the Thread may hang upon 45 degrees of the Limb, or upon 1, or the first prick of the Quadrat, and the distance from the foot of the Tower will be equal to the height of the Tower a∣bove the eye, which accordingly measure, and thereto add the height of the eye above the ground, and you will have the Altitude of the Tower.

So if I should stand at D and find the Thread to hang over 45^d of the Limb, I might conclude the distance between my Station and the Tower to be equal to the height of the Tower above my eye, and thence measuring it find to be 96 yards, so much would be the height of the Tower above the eye.

If I remove farther in till the Thread hang upon the

• second point of the Quadrat, then will the Altitude of the Tower above the level of the eye be
• third point of the Quadrat, then will the Altitude of the Tower above the level of the eye be
• fourth point of the Quadrat, then will the Altitude of the Tower above the level of the eye be
• fifth point of the Quadrat, then will the Altitude of the Tower above the level of the eye be

• twice
• thrice
• four times as much as the distance from the Tower is to the Station.
• five times as much as the distance from the Tower is to the Station.

So removing to C; I find the Thread to hang upon the second point of the Quadrat, and measuring the distance of that Station from the Tower, I find it to be 48 yards, whence I may conclude the Tower is twice as high above my eye, and that would be 96 yards.

So if I should remove so much back that the Thread should hang upon

• 2 of the shad∣dows &c. as E the distance between
• 3 of the shad∣dows &c. at F the distance between
• 4 of the shad∣dows &c.
• 5 of the shad∣dows &c.

the Station and the foot of the Tower would be

• twice
• thrice
• four times
• five times

as much as the height of the Tower above the eye, and consequently

if I should measure the distance between D and E where it hung upon 1 and 2 of the Shaddows, or between E and F, where it hung upon 2 and 3 of the shaddows, &c I should find it to be equal to the Altitude; but other ways of doing it when inaccessible will afterwards follow.

A Second way at one Station.

[illustration]

WIth any dimension whatsoever of a competent length, mea∣sure off from the foot of the Object, whether Tower or Tree, just 10 or 100, &c. of the said Dimensions, as suppose from B to K, I measure of an hundred yards; there look through the sights of the Quadrant to the Top of the Object at A, and what parts the Thread hangs upon in the Quadrat or Shadows, shews the Altitude of the Object in the said measured parts, and so at the said Station at K the Thread will hang upon 96 parts, shewing the Altitude A B to be 96 yards above the Level of the eye, and so if any other parts were measured off, they are to be multiplyed by the Tangent of the Altitude, or parts cut by the Thread, rejecting the Ciphers of the Radius, as in the next Proposition.

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A third way by a Station at Random.

[illustration]

TAke any Station at Random as at L, and looking through the sights observe upon what parts of the Quadrat or Shadows the Thread falls upon, and then measure the distance between the Sta∣tion and the foot of the object, and the Proportion will hold.

• As the Radius
• To the Tangent of the Altitude, or to the parts cut in the Quadrat or shadows,
• So is the distance between the Station and the Object
• To the height of the Object above the eye.

So standing at L, the Thread hung upon 30^d 58^m of the Limb, as also upon 600 of the Quadrat, the Tangent of the said Ark and the measured distance L B was 160 yards, now then to work the former Proportion, multiply the distance by the parts of the Qua∣drat, and from the right hand of the product cut off three places, and you have the Altitude sought.

〈 math 〉〈 math 〉

In this Work the Radius or Tangent of 45^d is assumed to be 1000.

To measure part of an Altitude, as suppose from a window in a Tower to the top of the Tower, may be inferred from what hath been already said; first get the top of the Tower by some of the

former ways, and then the height of the window which substract from the former Altitude and the remainder is the desired distance between the window and the top of the Tower.

The former Proportion may also be inverted for finding of a distance by the height, or apprehending the Tower to lye flat on the ground, and so the height to be changed into a distance and the distance into a height, the same Rules will serve only the height of a Tower being measured, and from the top looking to the Ob∣ject through the sights of the Quadrant, what Angle the Thread hangs upon is to be accounted from the right edge of the Quadrant towards the left; but in taking out the Tangent of this Ark, after I have observed it, the Thread must be laid over the like Ark of the Quadrant from the left edge towards the right, and from the Qua∣drat or Shaddows the Tangent taken out by the Intersection of the Thread, and so to measure part of a distance must be done by get∣ting the distances of both places first, and then substract the lesser from the greater.

To find the Altitude of any Perpendicular, by the length of its shadow.

THis will be like the first Proposition, with the Quadrant take the Altitude of the Sun, if in so doing the Thread hang over the

• 1^st pricks in the Quadrat, the length of the Shaddow is equal to the height of the Object Tree, or Perpendicular
• 2^d pricks in the Quadrat, the length of the Shaddow is double the height of the Object Tree, or Perpendicular
• 3^d pricks in the Quadrat, the length of the Shaddow is triple the height of the Object Tree, or Perpendicular
• 4^th pricks in the Quadrat, the length of the Shaddow is four times the height of the Object Tree, or Perpendicular
• 5^th pricks in the Quadrat, the length of the Shaddow is five times the height of the Object Tree, or Perpendicular

whatever it be; But if it hang

• 1 in the shaddows
• 2 in the shaddows
• 3 in the shaddows
• 4 in the shaddows
• 5 in the shaddows

the highth of the object is

• equal to the length of the
• double the length of the
• triple the length of the
• four times the length of the
• five times the length of the

Shaddow which may happen where the Sun hath much Altitude, as in small Latitudes, and so the length of the Shadow being forth∣with measured, the height of the Gnomon may be easily attained.

If the Thread in observing the Altitude hang on any odd parts of the Quadrant or Shadows, the Proportion will hold as before.

• As the Radius
• To the length of the Shadow,
• So the Tangent of the Suns Altitude, or the parts cut by the Thread
• To the height of the Gnomon to be wrought as in the third Proposition;

if the length of the Gnomon, and the length of its shadow were gi∣ven; without a Quadrant we might obtain the Suns Altitude, for

• As the length of the Gnomon
• Is to the Radius
• So is the Length of its Shadow
• To the Tangent of the Complement of the Suns Altitude.

And the height of the Sun, and the length of the Gnomon as∣signed; We may find the length of the Shadow by inverting the Proportion aforesaid.

• As the Radius
• To the length of the Gnomon
• So the Cotangent of the Suns Altitude
• To the Length of the Shadow.

To find an innaccessible Height at two Stations.

Assume a Station any where, as at G, and there observe the Altitude of the Object A B which admit to be 70^d Now it is

well observed by Mr. Phillips, that if you remove so far back as that the Object may appear but half so high, as suppose at I it ap∣pears to have but 35^d of Altitude that then the distance between

[illustration]

these two Stations G and I, is equal to the length of the Hipotenu Sall, or scaling Ladder A G; and this must needs be so, because the Acute Angle A G B being the Complement of the Obtuse An∣gle at G is equal to the Sum of the other two Acute Angles G A I and G I A, which Sum is likewise Complement of the Obtuse Angle at G to 180, but these Angles by Supposition being equal each to other will subtend equal Sides; Admit then I measure the distance G I, and find it to be 102 yards and a tenth, the Proportion to get the Altitude will hold

• As the Radius
• To the said measured distance
• So is the Sine of the Angle at the nearest Station
• To the Altitude of the Object.

To work this with the Pen, Out of the Line of Sines take the Sine of 70^d with Compasses, and measure it on the equal parts, where admit it reach to 94 parts,

Then multiply the said number by the distance 102, 1 and the Product will be 95, 974 from which cutting off three figures to the right hand the residue being 95, 974, is the Altitude sought feré, but should be 96 caused by omitting some fractionate parts in the distance which we would not trouble the Reader withall.

Another more general way by any two Stations taken at randome.

[illustration]

ADmit the first Station to be as before at G where the obser∣ved Altitude of the Object was 70^d and from thence at plea∣sure I remove to H, where observing again I find the Object to appear at 48^d 29^m of Altitude, and the measured distance between G and H to be ••0 yards, a general Proportion to come by the Al∣titude in this case will hold.

• As the difference of the Cotangents of the Arks cut at either Sta∣tion.
• Is to the Distance between the two Stations,
• So is the Radius
• To the Altitude of the Object or Tower.

To save the substracting of the two Arks from 90^d to get their Complements, I might have accounted them when they were ob∣served from the right edge of the Quadrant towards the left, and have found them to have been 20^d and 41^d 31^m; to work this Pro∣portion lay the Thread over these two Arks in the Limb from the left edge towards the right, and take out their Tangents out of the Quadrat and Shadows, then substract the less from the greater, the remainder is the first tearm of the Proportion, being the Divisor in the Rule of three to be wrought by annexing the Ciphers of the Radius to the Distance; or as Multiplication in Decimalls; and then dividing by the first tearm the Quotient shews the Altitude sought,

• ...

• 41^d 31′ Tangent from the Shadows is—885
• 20^d Tangent from the Quadrat is—364
• Difference—521

By which if I divide 50000— the distance encreased by an∣nexing the Ciphers of the Radius thereto, the Quotient will be 96 fere the Altitude sought.

This may be performed otherwise without the Pen, as shall af∣terwards be shewn;

If the distance from the Stations either to the foot or the top of the Tower be desired, the Proportions to Calculate them will be

• ...As the difference of the Cotangents of the Arks cut at either Station
• ...Is to the distance between those Stations,
• ...So is the Cotangent of the
  • greater Arke to the lesser
  • lesser Arke to the greater

distance from the Station to the foot of the Tower:

And so is the Cosecant of the

• greater Ark to the lesser distance
• lesser Ark to the greater distance

from the top of the Tower to the eye,

Or having first obtained the Height, we may shun these Secants, for

• As the Sine of the Ark of the Towers observed Altitude at the first Station.
• Is to the height of the Tower or Object above the eye,
• So is the Radius
• ...To the distance between the Eye and the Top of the Tower;

and by the same Proportion using the Ark at the second Station, the distance thence between the Eye and the Top of the Tower may be likewise found.

If any one desire to shun this Proportion in a difference, as per∣haps wanting natural Tables, it may be done at two Operations: The first to get the distance from the eye to the Top of the Tower at first Station, in regard the difference of two Arks is equal to the

difference of their Complements, it will hold

• As the Sine of the Ark of difference between the Angles observed at each Station
• Is to the distance between the two Stations,
• So is the Sine of the Angle observed at the furthest Station,
• To the distance between the eye and the Top of the Object at the first Station,

which being had, it then holds,

• As the Radius
• To the said Distance,
• So the Sine of the Angle observed at the first Station,
• To the Altitude of the Object.

The rest of the Lines on the Quadrant are either for working of Proportions, or for Protractions, and Dyalling depending thereon; wherefore I thought fit to reduce all the common Cases of Plain and Sphoerical Triangles to setled Cannons, and to let them precede their Application to this or other Instruments, which shall be endea∣voured; it being the cheif aim of this Book to render Calculation facil, in shunning that measure of Triangular knowledge hitherto required, and to keep a check upon it.