Mill B, and the Ciftle D ftand from the Water-fide. For if you take the distance betwixt any two of the places, with your Compaffes, and try it upon the fame Scale that you laid down your Stationary diftances, gives you the diftance required. Prob. 2. To find the distance between any two Places, both removed from the Obferver. Fig. 69. Let the two places be B and D, and let their diftances be required by an Obferver ftanding at C, 1. Let the Angle BCG be taken, between one of the Places, as B, and any vifible mark; fuppofe G ftanding about the the middle of the diftance, and likewise let the Angle GCD be taken. 2. Then leaving a visible mark at C, let the Obferver go backwards into another Station as at H, in fuch manner, as that he being at H might fee the mark C and G, in a right Line; and let him measure the diftance between the two Stations Cand H. 3- At H let him take the Angle BHG, and GHD, as he did before at C. This being done in the Triangle HBC, because theoutward Angle BCG is equal to both the inward and oppofite Angles BHC and HBC, therefore by Sabftracting BHC out of BCG, there will remain the Angle HBC. Thus all the Angles in the Triangle HBC, and the fide HC being given either of the other two fides are found by the third Cafe of Plane Triangles. Again, in the Triangle HDC, the Angle HDC, and either of the fides CD or HD may be found in the fame manner. Lastly, In the Triangle BCD, the two Sides BC and DC being found (as is already taught) and the Argle BCD, by Obfervation, the other two Angles CBD and CDB, may be found by the fourth Cafe of Plane Triangles, and confequently the Side BD by the third Cafe of PlaneTriangles, which is the distance required. Of Levelling, or Meafuring the Inequallity of Places, as to their Heights. To Let Fig. O find out the difference of heights 70.. of one place from another, in the rifing and falling, which is of conftant ufe in conveying of Water, either above the Ground for Fountains, &c. or under the Ground for Adyts or Soughs, &c. your Inftrument be carefully made, whe ther it be a Quadrant, Water-level, or any other; the beft I Account to be a brafs T, the fights upon the top of the T, to be Perfpective Glaffes, which must be tried before ufed, and the Glaffes are to ftand always one way, this will endure longer Stations. than than ordinary, and is for many reafons the Beft, it fubftantially made, and there muft be two mark Boards placed upon quarter Pikes, that your Affiftants may lift them higher or lower, as they fhall be directed. Then fet the Level as near as you can betwixt the 2 Marks which your Affiftants hold upright in their Hands, with the flipping Marks; turning to one, caufe him to Hold or fet the flit and Black ftroke even with the Level fights, and fo the other. The difference of thefe fights, in Inches and tenth parts gives the Afcent or Defcent, and this is for one fimple Station; but if it requires both Afcents and Defcents, 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 otherwife, higher. An Example will clear all. I am to give the difference of heights of the places A and B, from the Line of the Level SB, chufing my fift Station at C, where I plant my Inftrument, betwit, the Quater Pikes A and F, and fetting my Level firm, the Affiftan's litting up and down the mark Boards till both ways the fights take the Black ftrokes at D and E, in a little Table made, fet down the heights of thofe ftroakes from the Ground, in two Columns, one for the left Hand, the other for the Right, as you fee in the Table adjoining, wherein AD (for the left band) is found to be four Feet three Inches of an Inch, and EF (for the right Hand) feven Feet, 1.5. Again, Let the Second Station be at G, and the left Hand height FH be ten Feet, three Inches and a half, and the Right-hand height IK three Feet, 3.7 Inches. Again, The third Station let be at M, and the Height IL 2 feet, 9,4 Inches, and ON 12 feet, 1. 5. Inches. Laftly, Let the fourth Station be at P, and the height OQ three Feet 10, 9 Inches, and BR 11 Feet 9, 8Inches. The Sun of the heights on the Left hand is 212 Feet and three Inches, that of thofe on the Right 34 Feet and 4,5 Inches; their difference is 13 Feet and 1,5 Inches and fo much is B lower then A. Sum of the Hights on the Right-hand Their Differ. And fo much is B lower then A. The Ufe of the Line of Proportion, or Numbers commonly called Gunter's Line. THE Irg nious Mc Gunter and several others have fufficiently handled this Subject, therefore I might have faved my felf that trouble, but because it will be expect ed here, and the Book more ufeful, I fhall fay fomthing to that purpose, and begin first with, Numeration upon the Line. Numeration by the Line may be under ftood from this one Thought, viz. That what Denomination foever the fift at the beginning of the Line is, that in the middle will be ten times, and that at the end will be an hundred times fo many: Which if underflood, it will not be difficult to know what the intermediate Figures and parts are. Example I. To find the place of 25, you may call' the 1 at the beginning of the Line, but then will that in the middle be 10, and the two which ftands upon the 2d Ra |