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Mill B, and the Castle D ftand from the Water-lide. For if you take the distance bitwixt any two of the places, with your Compasses, and try it upon the same Scale that you laid down your Sationary distances, gives you the distance required.
To find the distance between any two Places, both removed from the Observer.
Fig. 69. Let the two places be B and D, and let their diftances be required by an Observer standing at C,
1. Let the Angle BCG be taken, b:tween one of the Places, as B, and any vi. fible mark; suppose G standing about the the middle of the distance, and likewise ler the Angle GCD be taken.
2. Then leaving a visible mark at C let the Observer go backwards into another Station as at H, in such manner, as that he being at H might see the mark'C and G, in a right Line ; and let him measure the diftarce between the two Stations Cand H. 3.
At H let himn take the Angle BHG, and GHD, as he did before at C. This being done in the T iangle HBC, because the ourward Angle BCG is equal to both the inward and oppofire Angles BHC and HBC, therefore by Sabstracting BHC out of BCG, there will remain the Angle HBC. Thus all the Angles in the Triangle HBC, and
the side HC being given either of the other two sides are found by the third Case of Plane Triangles.
Aguin, in the Triangle HDC, the Angle HDC, and either of the fides CD or HD may be found in the same manner.
Lastly, In the Triangle BCD, the two Sides BC and DC being found (as is 'already taught) and the Argle BCD, by Observation, the other two Angles CBD and • CDB, may be found hy the fourth Cafe of Plahe Triangles, and consequently the Side BD by the third Case of Plan:Triangles, which is the distance required.
Of Levelling, or Measuring the Inequallity of Places, as to their Heights.
o find out the difference of heights 70., of one place from another, in the risicg and falling, which is of constant ufe in conveying of Water, either above the Ground for Fountains, C. or under the Ground for Adyts or Soughs, &c. Let your Instrument be carefully made, wheiher it be a Quadrant, Water-level, or anyother; the best I Account to be a brass T, the fights upon the top of the T, to be Perspective Glaffes, which must be iried before used, and the Glasses are to stand always one way, this will endure longer Stations
than ordinary, and is for inany reasons the B:st, it fubftantially made, and there muft be iwo mark Boards placed upon quarter Pikes, that your Afiftants may lift them higher or lower, as they shall be directed.
Then set the Level as near as you can betwixt the 2 Marks which your Afiftants hold upright io their Hands, with the flipping Marks; turning to one, cause him to Hold or set the fit and Black ftroke even with the Level lights, and so the other. The difference of these fights, in Inches and tenih parts gives the Ascent or Descent, and this is for one fimple Station ; but if it requires both Ascents and Descents, then in a Book fet down your Bark Stations in one Column and you fore Stations in another, Sum up both the Columns, and take the difference of them; if they be equal, the two placesare Level, if your foreStation exceed, the difference is lower, if otherwise, higher. An Example will clear all. I am to give the diffirence of heights of the places A and B, from the Line of the Level SB, chusing my fiift Station at C, where I plant iny Inftrument, betwit, the Quater Pikes A and F, and setting my Level firm, the Allifanis litung up and down the mark Boards till both ways the fights take the Black strokes at D and E; in a little Table made, set down the heights of those stroakes from the Ground, in iwo Columns, one for the left
Hand, the other for the Right, as you see in the Table adjoining, wherein AD (for tbe left band) is found to be four Feet three Inches of an Inch, and EF (for the right Hand) seven Feet, 1.5.
Again, Let the Second S-ation be at G, and ihe left Hand height FH be sen Feet, three Inches and a half, and the Right-hand height I K three Feet, 3.7 Inches. Again, The third Station let be at M, and the Height IL 2 feet, 9,4 Inches, and ON 1 2 teet, 1. s. Inches. Lastly, Lot the fourth Station be at P, and the height OQ three Feet 10, 9 Inches,
9 and BR 11 Feet 9, 8Inches. The Suin of the heights on the Left hand is 212 Feet and three Inches, that of those on the Right 34 Feet and 4; 5. Inches; their difference is 13 Feet and i, s Inches and so much is B lower then A.
Stations. (Heights on the Heights on the
Left-hand. Right hand.
Sum of the Highes on the Right hand 34. 4 5
The Use of the Line of Proportion, or
Numbers commonly called Gunter's
HE Irg'nious Mr Gunter and several
others have sufficiently handled this Subject, therefore I might have saved my self that trouble, but because it will be expect ed here, and the Book more useful, I Mall fay somthirg to that purpose, and begin first with,
Numeration upon the Line.
Numeration by the Line may be under stood from this one Thought, viz. That what Denomination foever the first at the beginning of the Line is, that in the middle will be ten times, and that at the end will be an hundred tiines fo many : Which if understood, it will not be difficult to know what the intermediate Figures and parts
To find the place of 25, you may call the I at the beginning of the Line, but 1, then will that in the iniddle be 10, and the two which ftands, upon the 2d