hydrostatics: OR, INSTRUCTIONS CONCERNING WATER WORKS. COLLECTED Out of the Papers OF Sir SAMVEL MORLAND. CONTAINING The Method which he made use of in this Curious Art. LONDON: Printed for John Laurence at the Angel in the Poultry, over ●gainst the Compter. 1697. TO THE READER. THE following Tables I Received from Sir Samuel Morland, amongst the rest of his Mathematical Papers, all of which Kind he was pleased to bestow on me not long before his Death. As for these which I now Publish, he told me particularly, that they contained the Mystery of that Art and nimble Dispatch, which he was Master of, in the Making and Managing of (more especially) such Mechanical Engines as relate to the Water; in the Improvement of which sort he was so much happier than the rest of Mankind. He thought that it might be an acceptable and useful piece of Service to the World, to range these Materials in good Order; and where there should be occasion, to add so much light as might make them easily Intelligible to a Common Reader. How I should do this, he gave me large Directions from his own Mouth, and I have punctually observed them, in the Completing of this Piece; so that here are plain and easy Rules and Directions delivered in a perspicuous manner, that guide the Practitioner into the Concisest way of Calculation in these Matters; and almost infallibly secure him from Mistakes and Errors, which are so vexatious and expensive. And I think it is not necessary to give any larger Account of this Treatise. What other of his Papers may hereafter be made Public, must be left to further Enquiry and Consideration. Joseph Morland. THE FIRST TABLE. A Table of Square Roots of all Numbers from 1 to 100 Squ. Number Squa. Roots. Difference 1 1.00 0.41 2 1.41 0.32 3 1.73 0.27 4 2.00 0.24 5 2.24 0.21 6 2.45 0.20 7 2.65 0.18 8 2.83 0.17 9 3.00 0.16 10 3.16 0.16 11 3.32 0.14 12 3.46 0.15 13 3.61 0.13 14 3.74 0.13 15 3.87 0.13 16 4.00 0.12 17 4.12 0.12 18 4.24 0.12 19 4.36 0.11 20 4.47 0.11 21 4.58 0.11 22 4.69 0.11 23 4.80 0.10 24 4.90 0.10 25 5.00 0.10 26 5.10 0.10 27 5.20 0.09 28 5.29 0.10 29 5.39 0.09 30 5.48 0.09 31 5.57 0.09 32 5.66 0.08 33 5.74 0.09 34 5.83 0.09 35 5.92 0.08 36 6.00 0.08 37 6.08 0.08 38 6.16 0.08 39 6.24 0.08 40 6.32 0.08 41 6.40 0.08 42 6.48 0.08 43 6.56 0.07 44 6.63 0.08 45 6.71 0.07 46 6.78 0.08 47 6.86 0.07 48 6.93 0.07 49 7.00 0.07 50 7.07 0.07 51 7.14 0.07 52 7.21 0.07 53 7.28 0.07 54 7.35 0.07 55 7.42 0.06 56 7.48 0.07 57 7.55 0.07 58 7.62 0.06 59 7.68 0.07 60 7.75 0.06 61 7.81 0.06 62 7.87 0.07 63 7.94 0.06 64 8.00 0.06 65 8.06 0.06 66 8.12 0.07 67 8.19 0.06 68 8.25 0.06 69 8.31 0.06 70 8.37 0.06 71 8.43 0.06 72 8.49 0.05 73 8.54 0.06 74 8.60 0.06 75 8.66 0.06 76 8.72 0.05 77 8.77 0.06 78 8.83 0.06 79 8.87 0.05 80 8.94 0.06 81 9.00 0.06 82 9.06 0.05 83 9.11 0.06 84 9.17 0.05 85 9.22 0.05 86 9.27 0.06 87 9.33 0.05 88 9.38 0.05 89 9.43 0.06 90 9.49 0.05 91 9.54 0.05 92 9.59 0.05 93 9.64 0.06 94 9.70 0.05 95 9.75 0.05 96 9.80 0.05 97 9.85 0.05 98 9.90 0.05 99 9.95 0.05 100 10.00 The Use of the foregoing TABLE. To find the Square Root of any given Number as far as Three Figures. DIstinguish the given Number by Points, as is usual in the Extracting of Square Roots, and observe how many Figures belong to the first Point, which will be either one or two: If only one, then seek that Figure in the foregoing Table of Square Roots, in the first Column from one to nine inclusive, and write out the Root standing directly over-against it in the second Column; and take also the next lower difference out of the third Column, which you Multiply by the two next Figures of your given Number, and from the Product cut off two Figures, and add the remainder to the Root first written out. The Sum is the desired Root. But if there be two Figures belonging to the first Point, then seek them in the first Column from 10 to 99 inclusive, and proceed as before. Example. Extract the Square Root out of 276438. the Number distinguished by the Points standeth thus; 27̣64̣38̣. and the Figures belonging to the first Point are 27. which being looked in the first Column, you find over against it in the second Column 520. and in the third Column the next lower difference 10, which Multiplied by 64 (the two next following Figures in the given Number) the Product is 640. and cutting off two Figures, the remainder is 6. to be added to 520. and the desired Root is 526. THE SECOND TABLE. A TABLE of Cube Roots, from 1 to 10, and then continued for every Tenth Number from 10 to 100 Number Cube Roots Difference 1 1.00 0.26 2 1.26 0.18 3 1.44 0.15 4 1.59 0.12 5 1.71 0.11 6 1.82 0.09 7 1.91 0.09 8 2.00 0.08 9 2.08 0.07 10 2.15 0.56 20 2.71 0.40 30 3.11 0.31 40 3.42 0.26 50 3.68 0.23 60 3.91 0.21 70 4.12 0.19 80 4.31 0.17 90 4.48 0.16 100 4.64 0.15 110 4.79 0.14 120 4.93 0.14 130 5.07 0.12 140 5.19 0.12 150 5.31 0.12 160 5.43 0.11 170 5.54 0.11 180 5.65 0.10 190 5.75 0.10 200 5.85 0.09 210 5.94 0.09 220 6.04 0.09 230 6.13 0.08 240 6.21 0.09 250 6.30 0.08 260 6.38 0.08 270 6.46 0.08 280 6.54 0.08 290 6.62 0.07 300 6.69 0.07 310 6.77 0.07 320 6.84 0.07 330 6.91 0.07 340 6.98 0.07 350 7.05 0.06 360 7.11 0.07 370 7.18 0.06 380 7.24 0.07 390 7.31 0.06 400 7.37 0.06 410 7.43 0.06 420 7.49 0.06 430 7.55 0.06 440 7.61 0.05 450 7.66 0.06 460 7.72 0.05 470 7.77 0.06 480 7.83 0.05 490 7.88 0.06 500 7.94 0.05 510 7.99 0.05 520 8.04 0.05 530 8.09 0.05 540 8.14 0.05 550 8.19 0.05 560 8.24 0.05 570 8.29 0.05 580 8.34 0.05 590 8.39 0.04 600 8.43 0.05 610 8.48 0.05 620 8.53 0.04 630 8.57 0.05 640 8.62 0.04 650 8.66 0.05 660 8.71 0.04 670 8.75 0.04 680 8.79 0.05 690 8.84 0.05 700 8.89 0.03 710 8.92 0.04 720 8.96 0.04 730 9.00 0.04 740 9.04 0.05 750 9.09 0.04 760 9.13 0.04 770 9.17 0.04 780 9.21 0.03 790 9.24 0.04 800 9.28 0.05 810 9.33 0.03 820 9.36 0.04 830 9.40 0.04 840 9.44 0.03 850 9.47 0.04 860 9.51 0.04 870 9.55 0.03 880 9.58 0.04 890 9.62 0.03 900 9.65 0.04 910 9.69 0.04 920 9.73 0.03 930 9.76 0.04 940 9.80 0.03 950 9.83 0.03 960 9.86 0.04 970 9.90 0.03 980 9.93 0.04 990 9.97 0.03 1000 10.00 The Use of the Second TABLE. To find the Cubick Root of any given Number as far as Three Figures. DIstinguish the given Number by Points, as is usual in Extracting Cubick Roots. Then to the first Point towards the left Hand, there will belong either one Figure, or two, or three Figures. If it be one, then look the same in the Table of Cubick Roots, from 1 to 9 inclusive, if there be two Figures belonging to the first Point, then look the first of them from 10 to 90 inclusive, if there be three, look the two first from 100 to 990 inclusive, and write out the Root standing directly over-against it in the Second Column, and the next lower Difference out of the third Column, which you Multiply by the two next Figures of your given Number, and from the Product cut off two Figures, and add the remainder to the Root first written out: The Sum is the desired Root. Example. Extract the Cubick Root out of 34̣167̣942̣. The Number distinguished by Points standeth thus: 34167942. and the Figures belonging to the first Point are 34, whereof the first, viz. 3. is to be looked between 10 and 90. inclusive; and you find 30. and the Root of this in the Second Column 311. to be written out, and the next lesser Difference in the third Column is 31. which Multiplied by 41. (the two next following Figures in the given Number) the Product is 1271. and cutting off two Figures, the remainder is 12 or 13. (because the Figures cut off, viz. 71. are more than the half of 100) which added to 311. (the Root first written out) the Sum is 324. the desired Cubick Root. THE THIRD TABLE. A TABLE of Cube-Root-Inches, from a Pint to a Gallon, from a Gallon to a Barrel, from a Barrel to a Tun, from One Tun to Seventy, and from thence by Decimal Steps to 10000 Pints Cube Root Inch. 1 3.27 2 4.16 3 4.72 4 5.20 5 5.60 6 5.95 7 6.27 Firkins Gallons Cube Root Inch. 1 6.55 2 8.26 3 9.45 4 10.40 5 11.21 6 11.91 7 12.54 8 13.11 1) 9 13.63 10 14.12 11 14.58 12 15.01 13 15.41 14 15.80 15 16.17 16 16.52 17 16.86 2) 18 17.18 19 17.49 20 17.80 21 18.09 22 18.37 23 18.64 24 18.91 25 19.17 26 19.42 3) 27 19.67 28 19.91 29 20.14 30 20.37 31 20.60 32 20.81 33 21.03 34 21.24 35 21.45 36 21.65 Cube-Root-Inches of Barrels and Tuns. Tuns Barrels Cube Root Inch. 1 21.65 2 27.28 3 31.22 4 34.37 5 37.02 1) 6 39.34 7 41.42 8 43.31 9 45.03 10 46.65 11 48.15 2) 12 49.57 13 50.91 14 52.18 15 53.40 16 54.56 17 55.67 3) 18 56.74 19 57.77 20 58.77 21 59.73 22 60.67 23 61.57 4) 24 62.45 25 63.31 26 64.14 27 65.75 28 65.95 29 66.52 5) 30 67.28 31 68.01 32 68.74 33 69.45 34 70.14 35 70.82 6) 36 71.49 37 72.15 38 72.79 39 73.42 40 74.05 41 74.66 7) 42 75.26 43 75.86 44 76.44 45 77.01 46 77.58 47 78.14 8) 48 78.69 Cube-Root-Inches of Tuns. Tuns Cube Root Inch. 9 81.84 10 84.76 11 87.50 12 90.07 13 92.51 14 94.82 15 97.03 16 99.14 17 101.16 18 103.11 19 104.95 20 106.80 21 108.55 22 110.24 23 111.89 24 113.49 25 115.04 26 116.56 27 118.03 28 119.47 29 120.88 30 122.25 31 123.60 32 124.91 33 126.20 34 127.46 35 128.70 36 129.91 37 131.10 38 132.28 39 133.40 40 134.56 41 135.67 42 136.76 43 137.84 44 138.90 45 139.94 46 140.97 47 141.99 48 142.99 49 143.97 50 144.95 51 145.91 52 146.86 53 147.80 54 148.71 55 149.63 56 150.53 57 151.42 58 152.30 59 153.17 60 154.03 61 154.88 62 155.73 63 156.56 64 157.38 65 158.20 66 159.01 67 159.80 68 160.60 69 161.38 70 162.15 80 169.53 90 176.32 100 182.62 200 230.09 300 263.39 400 289.90 500 312.28 600 331.85 700 349.35 800 365.25 900 379.88 1000 393.46 2000 495.73 3000 567.47 4000 624.58 5000 672.81 6000 714.96 7000 752.66 8000 768.92 9000 818.43 10000 847.68 20000 1068.06 30000 1222.57 40000 1345.61 50000 1449.52 60000 1540.34 70000 1621.56 80000 1695.37 90000 1763.25 CM. 1826.28 THE Use of this third Table is so Obvious to every Man's Capacity, that it needs no Explanation; for if there be occasion to make any Vessel in a Cubical Form, of which the Content aught to be a Pint, a Quart, a Gallon, etc. This Table gives the Cubick Roots of their Respective Dimensions. THE FOURTH TABLE. A TABLE showing the true Content of Cubick Feet, (from 1 to 5) in Gallons and Cubick Inches, and (from 5 to 1000) in Barrels, Gallons, and Cubick Inches. Cubick Feet Barrels Gallons Cubick Inches 1 0 6 36 2 0 12 72 3 0 18 108 4 0 24 144 5 0 30 180 6 1 00 216 7 1 6 252 8 1 13 6 9 1 19 42 10 1 25 78 11 1 31 114 12 2 1 150 13 2 7 186 14 2 13 222 15 2 19 258 16 2 26 12 17 2 32 48 18 3 2 84 19 3 8 120 20 3 14 156 30 5 3 234 40 6 29 30 50 8 18 108 60 10 7 186 70 11 32 264 80 13 22 60 90 15 11 138 100 17 00 216 200 34 1 150 300 51 2 84 400 68 3 18 500 85 3 234 600 102 4 168 700 119 5 102 800 136 6 36 900 153 6 252 1000 170 7 186 The several Uses of the Fourth TABLE. First Use. ONE Use of this Table is this, viz. either by having given the Dimensions of any Rectangular Vessel, whose sides are Plain, to find the true Content in Gallons, Barrels, etc. or else, the Content of any such Vessel being given with one of its three Dimensions, to determine the two remaining Dimensions. First, Let it be required to know the true Content of a Rectangular Vessel, of which the length is seven, the breadth three, and the height five Feet. The Answer is this: 7 into 3 into 5, make 105 Cubick Feet: Now by this Table I find that 100 Cubick Feet contain 17 Barr. 216. Cub. In. and by the same 5 Cubick Feet, contain 30 Gall. 180 Cub. In. which two Sums being added together make 17 Bar. 31 Gall. 114 Cub. In. which is the Content of the aforesaid Vessel. Again, The Content of a Vessel being given (viz. four Barrels) and the length of that Vessel (six Feet) let it be required to find the Breadth and Depth. Answer, By this Table I find that six Cubick Feet contain one Barrel, and 216 Cubick Inches. By this I know the Content of a Vessel, whose Length is six Feet, its Breadth one, and its Height one; therefore two such Vessels joined together upon a Plane, or (which is all one) a Vessel six Feet long, two Feet wide, and one Foot deep, contains two Barrels, one Gallon, and one Hundred and fifty Cubick Inches, and by Consequence a Vessel six Feet long, two Feet wide, and two Feet deep, contains four Barrels, three Gallons, and eighteen Cubick Inches, which is something over the given Content, but near enough for common use. And by this Method may any Cistern be designed near enough the Truth, with great Ease and Expedition. But if it be required to perform these or the like Operations more exactly, the following Method will guide the Practitioner, several Precognita being first laid down. 1. The Original of all long Measures is an Inch, whereof twelve make a Standard English Foot, 36 Inches make a Yard; 72 make a Fathom, 198 make a Perch, 7920 make a Furlong, 65360 make a Mile. 2. The Original of all Square Measures is a Square Inch, whereof 144 make a Square Foot, 1296 make a Square Yard, 39204 make a Square Rod, 6272640 make a Square Acre. 3. Of all Solid Measures the Original is a Cubick Inch, whereof 1728 make a Cubick Foot, 15,552, make a Cubick Yard. Again 35 ⌊ 25 make a Pint, 70 ⌊ 5 make a Quart, 282 make a Gallon, 10,152 make a Barrel, or Thirty six Gallons; 60,912 make a Tun or Six Barrels. 4. Any one Number being Multiplied either by itself, or any other Number, and that Product Multiplied by any third Number make a Solid, or the Content of a Rectangular Cistern, whose sides are plain. These Precognita being laid down the Operations will be as follow: For Example, The Content of a Cistern being given, viz. Four Barrels, and the length of that Cistern six Feet: Let it be required to find out two Numbers, which being Multiplied one into another, and that Product into the given Length, make a Content equal to Four Barrels. Having first Reduced the given Terms to their least Denominations, the General Rule is this: The Content of any Rectangular Vessel being given together, with either the length, or breadth, or Depth of that Cistern: Divide the given Content by the given Dimension, and the Quotient by any Number less than itself: The last Divisor and Quotient are the two Terms sought. Thus having reduced the Four Barrels to 40,608 Inches, and the given 6 Feet to to 72 Inches, I divide the said 40,608 by the said 72. And again I divide the Quotient 564 Inches by any Number less than itself, suppose 40, the last Divisor 40, and its Quotient 14 ⌊ 1 are the two sides required. And after this Method may infinite Answers be given to this Question; so that the Operator in the second Division may from the given length choose either what Depth or Breadth he pleases, or from a given given Depth may choose what Length or Breadth he pleases. The Second Use of the Fourth TABLE. To Explain this Use it is necessary to premise this following Theorem: Like Solids are in proportion one to another, as the Cubes of their Hemologous' sides. Upon which Theorem depends this Problem: Having the Content of a Cistern, together with the Ratio of the Length, Breadth and Height given to find the sides. I. Example, Let it be demanded to frame a Cistern containing 1000 Cubick Feet, and the Ratio of the sides or three Dimensions, let be one two and four. First, I imagine, or frame in my mind, a Cistern, whose Length 4 Breadth 2 Height 1 The Content of it is 4×2×1 = (8) Therefore I say, 1. As the Content 8, is to the Content 1000, so is the Cube of the side 4. viz. the Cube 64. to the Cube 8000, whose Cube-Root is = 20. 2. As the said 8 to the said 1000 so is the Cube of the side 2. viz. the Cube 8. to the Cube 1000 whose Cube Root is = 10. 3. As the said 8 to the said 1000 so is the Cube of the side 1. viz. Cube 1. to the Cube 125. whose Cube Root is = 5. So then of the Cistern demanded to be framed, the Length 20 Breadth 10 Height 5 For 20×10×5 = 1000; and 1. As 1 to 2, so 5 to 10. 2. As 2 to 4, so 10 to 20. 3. As 1 to 4, so 5 to 20. II. Example, Let a Cistern be demanded, containing 600 Cubick Feet, and the Ratio of the sides, as 3, 4, 5. First, I frame in my mind a Cistern of 3 4 5, 3×4×5 = 60. wherefore. 1. As to the Content 60. to the Content 600. so is the Cube of the side 3. viz. the Cube 27 to the Cube 270. whose Cube Root is = 6. 463. 2. As the said 60 to the said 600. so is the Cube of the side 4. viz. the Cube 64, to the Cube 640, whose Cube Root is = 8. 617. 3. As the said 60 to the said 600. so is the Cube of the side 5. viz. the Cube 125. to the Cube 1250, whose Cube Root is = 10. 77 2. So than Content Length Breadth height Of the given Cistern 60 5 4 3. Of the Cistern demanded 600 10.772 8.6176 6 ⌊ 4632 For 6.4632 into 8.6176 into 10.7720 is = 599. 978. And 1. As 3 to 4, so is 6.4632 to 8.6176 2. As 4 to 5, so is 8.6176. to 10.7720 3. As 3 to 5, so is 6.4362 to 10.7720. This Problem being clearly Answered by the two foregoing Examples; in the first of which the Length, Breadth and Depth of the Cistern, that is required to be designed, are 20, 10 and 5; and its Content 1000 Cubick Feet; and in the second the Length, Breadth and Depth of the Cistern that is required to be designed are 10,7720, 8.6176 and 6.4632. and its Content 600 Cubcik Feet. The next thing to be done is to Convert their Contents into Gallons, Barrels, etc. which is to be done with much ease by this Fourth Table: For by that Table 1000 Cubick Feet (which is the Content of the first Cistern required to be designed) contain 170 Bar.— 7 Gall.— 186. Cub. Inch. And in the second Example the Content of the Cistern required to be designed, being 600 Cubick Feet; I find by the said Table, that 600 Cubick Feet contain 102 Bar.— 4 Gall.— 168 Cub. Inch. The Third Use. The third Use of this Table relates to Cylindrical Elliptical Vessels; for the better Explanation of which there are again several Precognita to be premised. I. Diana of any Circle periphery 1 3. 14159265 2 6. 28318530 3 9 42477795 4 12. 56637060 5 15. 70796325 6 18. 84955590 7 21. 99118455 8 25. 13274120 9 28. 27433385 II. Diana of any Circle Square Root of the Area 1 0. 88622692 2 1. 77245385. 3 2. 65868077 4 3. 54490770 5 4. 43114362 6 5. 31736155 7 6. 20358847 9 7. 97604231 If the Diana be (1) the Area is 0. 785398163. 4. Square any given Diameter, and then Multiply that Square by 0. 7853, etc. and the last Produce is the Area of the Circle. 5. The Length of an Ellipsis drawn into the Breadth, and that Product Multiplied by 0. 7853981, etc. gives the Area of an Ellipsis. 6. Circles in proportion to one another, as the Squares of their Diameters. First Problem. The Diameter of any Cilindrical Vessel being given, together with its Height, to find the true Content thereof in Gallons, Barrels, etc. For Example, Suppose in a Noblemen or Gentleman's Garden there be found a Basin, whose Diameter is 45 Feet, and its Depth 4 Feet, and it be required to know the true Content thereof in Gallons, Barrels and Tuns. Answer. By the Fourth Praecognitum, I square the given Diameter 45. and that Square I Multiply by the Fraction 0.7853, etc. and the Product 1590.232 is the Area of the Circle, which Multiply by 4, the Height of the Cylindrical Vessel, and the Product, viz. 6360.9— Cubick Feet, is the Content of the Basin. This being done, the next thing is to convert the said Cubick Feet into Gallons, Barrels and Tuns; which, by the help of the Fourth Table, is easily done. For by the said Table, 1000 Cubick Feet is equal to 170 Barr.— 7 Gall.— 186, Cub. In. which being Multiplied by Six gives 1020, Barr.— 42 Gall.— 1116, Cub. In. which is being reduced to its right Denomination, 1021 Barr.— 9 Gall.— 270 Cub. In. the Content of 6000 Cubick Inches. Again. 300 Cubick Feet is equal to 51 B.— 2 G.— 84. C. In. this being added to the foregoing Sum, viz. B. G. C. In. 1021 9 270 51 2 84 makes 1072 11 354 the Content of 6300 Cubick Feet. Lastly, 60 Cubick Feet contain 10 B.— 7 G.— 186 C. In. which being added to the Content of 6300 Cubick Feet, viz. B. G. C. In. 1072 11 354 10 7 186 makes 1082 18 540 which Reduced to its right Denomination is 1082 B.— 19 G.— 258 C. In. the Content of the Basin which was required. Second Problem. The long and short Diameters of an Elliptical Vessel being given, together with the Height, to find the true Content thereof in Gallons, Barrels, etc. Suppose in a Gentleman's or Nobleman's Garden there be an Elliptical Basin, whose Length is 45 Feet, whose Breadth is 35 Feet, and whose Depth is 4 Feet; and it be required to know the true Content thereof in Gallons, Barrels and Tuns. Answer. By the Fifth Precognitum, I Multiply 45 the Length into 35, the Breadth of the Basin; the Product which is 1575., I Multiply by the Fraction 0.7853, and the Product of these two Numbers Multiplied, will be 1236. 847. which I then Multiply by 4 the Depth, and thence arises 4947.3 ... Cubick Feet which is the Content of the Basin. Now I, as before, Convert the said Cubick Feet into Gallons, Barrels and Tuns, by the help of the Fourth Table. Thus, 1000 Cubick Feet is there equal to 170 Barr.— 7 Gall.— 186 Cub. In. which being Multiplied by 4 gives 680 Barr.— 28 Gall.— 744 Cub. In. which (being reduced to its right Denomination) is 680 Barr.— 30 Gall.— 180 Cub. In. Again, 900 Cubick Feet is equal to 153 Barr.— 6 Gall.— 252 Cub. In. this being added to the foregoing Sum, viz. Barr. Gall. Cub. In. 680 30 180 153 6 252 makes 833 36 432 which is (being reduced) 833 Barr.— 37 Gall.— 150 Cub. In. Lastly, 40 Cubick Feet contain 6 Barr.— 29 Gall.— 30 Cub. In. And 7 Cubick Feet contain 1 Barr.— 6 Gall.— 252 Cub. In. therefore 47 Cubick Feet contain 7 Barr.— 35 Gall.— 282 Cub. In. which being added to the Content of 4900 Cubick Feet, viz. Barr. Gall. Cub. In. 833 37 150 7 35 282 makes 840 73 150 which (being reduced to its right Denomination) is 842 Barr.— 1 Gall.— 150 Cub. In. the Content of the Basin, in Barrels, Gallons and Cubick Inches, which was required. But forasmuch as in the Practical part of hydrostatics, and the designing of Engines to raise Water to great Heights, by the means of Forcers; it will be often requisite to know the Contents and the Weight of less Cylinders, that is to say, whose Diameters are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 Inches, and the Length Indefinite (suppose from 1 to 100, or more). And because such Calculations are very tedious, I have here inserted the following Table. THE FIFTH TABLE Giving the true Content in Cubick Inches of Cylinders of different Diameters, from 1 to 12 Inclusive, and each of these Cylinders of a Foot or Twelve Inches in Height. Diana of Bases Solid Content in Cubick Inches. 1 9 ⌊ 424777961 2 37 ⌊ 699111843 3 84 ⌊ 823001647 4 150 ⌊ 79644736 5 235 ⌊ 61944900 6 339 ⌊ 29200656 7 461 ⌊ 81412004 8 603 ⌊ 18578044 9 763 ⌊ 40701476 10 942 ⌊ 47779600 11 1140 ⌊ 39813316 12 1357 ⌊ 16802624 As for the weight of Water, it is not to be determined absolutely, because almost all Wells, Springs and Rivers, are of a different weight, and therefore my advice is to all Engineers and Practitioners, to find out the exact weight of a Cubick Foot of that particular Water, which they have occasion to make use of; by which means they will easily discover the weight of any Cylinder of Water with these following Cautions. First Caution. When an Engineer desires to force up Water 50, or 100 Feet in Perpendicular Height, and designs to do this by a Forcer of 4, 5, or 6 Inches Diameter; but intends the Water shall be carried up the said 50, or 100 Feet in Perpendicular Height, through a Pipe of of 1 Inch and ½, 1 Inch and ¾, or 2 Inches and ¼ Diameter, the Water standing in any such perpendicular Pipe, is equivalent in weight to a Pipe of the same Perpendicular Height, whose Diameter is 4, 5, or 6 Inches, viz. the Diameter of the said Forcer; and indeed the less one of those Pipes is, the greater is the weight against the Forcer to raise up the Water in the same Moment or Interval of Time; that is to say, there is required more Weight to be laid upon that Forcer, to raise the Water through a Pipe of one Inch Diameter, than through a Pipe of four Inches Diameter. And whatsoever is here said of the weight of Water against a Foreer, in a Forcing Engine, is also true in Suction, by a Drawing Pump. For Example. A Pump whose Barrel or Pipe of Suction is four Inches Diameter; and the Pipe which reaches from the Barrel to the Water, through which it is drawn up but two Inches Diameter, requires more force or strength, than if it were drawn up through a Pipe of four Inches Diameter. For want of the Knowledge of this, many ignonorant Plumbers and Pump-makers, covet to draw their Water through less Pipes, which makes the Work more difficult: And though this seems to be a Paradox, yet 'tis a real truth; and the want of the right understanding thereof has occasioned very many great Mistakes by ignorant Practitioners. Second Caution. The true weight of Water in all Pipes, is to be determined by the Perpendicular Height of those Pipes. For Example. The weight of Water in Perpendicular Pipes of three Feet (in Height) and three Inches Diameter, is equal to the weight of Water contained in a Pipe of any Length whatever (be it a Rod, a Furlong, or more) which rises not more than three Feet above the Horrizontal Line; which seems likewise to be a Mystery, but is a real truth, as it lies in the Pipes; although if it be taken out and laid in a Balance, it will weigh one Hundred times as much or more, than the Water in the said Perpendicular Pipe. I must confess, that the Author had very small Encouragement to help our Engineers in things of this Nature, many of them having dealt very disingenuously with him; when he had, by near Forty Years Study and Practise, and the Expense of many a thousand Pounds, produced new and better ways of raising Water, than for aught I know, were ever known to former Ages, viz. by the means of 1. A Forcer moving up and down in a Chamber of Water, through a small Collar or Neck of Leather fastened in a Groove. 2. The Circular Motion of a Crank, reduced to a Perpendicular. 3. The Unequal Motion of a Crank exchanged for an Elliptical Equal Motion. Divers Persons have borrowed, some one part, some another, and making some small Alterations, call it their own Invention. This I am willing to let pass; and after all, to give them the following Items, to prevent their attempting Perpetual Motions, which most of them are apt to do by their Ignorance in hydrostatics; and not a few Gentlemen, in all Ages, have, by such vain imaginations of deceiving Nature, deceived themselves, and Ruined their Families. 1. As in Staticks a Pound weight suspended Perpendicularly, has a greater force than a Pound weight suspended on a rising Line in Proportion, as the Hypotenuse of a Rectangled Triangle, is longer than its Perpendicular; so in hydrostatics, if the two ends of a Syphon turned angular-wise, and a part of it filled with Water, or any other Liquor, be immerged in a Vessel of the same Liquor; that part which hangs Perpendicularly, shall be heavier than that which declines in proportion, as one side of that Syphon is longer than the other opposite side. 1. Fig: diagrams of an isosceles triangle with weights and a triangular siphon illustrating static and hydrostatic principles For Example. Let the two Triangles in Fig. 1. be Isosceles, and the two sides of the Triangle ABC (viz. AC and AB) equal. And so likewise the two sides of the Triangle DEF (viz. DE and OF) equal. In this case a Pound weight (P) and another (Q) are equally Ponderous; and so is the Water contained in the Syphon (FN) from the Superfices (GRACCUS) to (N) of an equal weight with the Water in the Syphon (DM) between the Superfices of the Water (GRACCUS) and (M) 2. Fig: diagram of 30-60-90 triangle with weights illustrating static principles But now in Fig. 2. because the side (AB) of the Triangle (ABC) has double the length of (BC) therefore a Pound weight (D) suspended Perpendicularly from (B) is Equiponderant to two Pounds (E and F) on the side (AB) according to the Doctrine of Statinks. And therefore an Horse drawing a weight of four Hundred Pound, upon an Ascent of thirty Deg. heaves at two Hundred Pound, which is one half, and the Ground bears the rest. 3. Fig. diagram of 30-60-90 triangular siphon illustrating hydrostatic principles And so in Fig. 3. because the side (DE) of the Triangle (DEF) has twice the Length of (OF) therefore the Water (HF) which hangs Perpendicularly, is equal in weight to ●he Water (DG) which has twice its Quantity and Length, according to the Doctrine of hydrostatics. 2. As in Staticks, a Pound weight on the one side of the Perpendicular Line makes an Equilibrium, with another Pound Equidistant from the Perpendicular Line on the other side: so in hydrostatics, one Tube of Water standing at any Height above the Horizontal Line, Equiponderates any other Tube o● Water that stands at the same Height, and is of the same Diameter. 3. As in Staticks if a les● weight raise a greater, it mus● be proportionably at a greate● distance from the Perpendicular Line, and have a greater Motion. So in hydrostatics, i● a less Tube of Water raise a greater Tube, it must be proportionably of a greater Length above the Horizontal Line than the other; and consequently the Descent of the Water in a lesser Tube, must have a greater Length than the Ascent of the Water in a greater Tube in proportion, as the Square of the Diameter of the greater Tube, exceeds the Square of the Diameter of the lesser Tube. And this length of Ascent and Descent of the lesser Tube of Water above the Horizontal Line, compared with the Ascent and Descent of the greater Tube, together with the proportion that the Square of the Diameter of the lesser Tube, bears to the Square of the Diameter of the greater Tube; answers exactly to the force and Motion of a Leaver, or rather of a lesser weight placed on a Balance at a greater distance from the Perpendicular to Counterpoise or raise a greater weight placed on the other side at a lesser distance according to the Doctrine of Staticks. Fig. 1. diagrams of unequal weights in balance and of U-shaped tubing with perpendicular angles, illustrating static and hydrostatic principles As in Fig. 1. A pound weight (A) is equiponderant to another (B) because equidistant from the Perpendicular (CD) So that part of the Tube (EL) whose Height above the Horizonal Line (FE) is of an equal weight with that part of the Tube (GM) which is of an equal leight above (FH) viz. (GH.) Fig. 2. diagrams of unequal weights and of a U-shaped tube with perpendicular angles with unequal diameters in its two vertical sections, illustrating static and hydrostatic principles In Fig. 2. as a pound weight A equiponderating three Pound (B, C, D,) must have thrice the Distance from the Perpendicucular (OF) and for every Foot or Inch (B, C, D,) ascends or descends A must ascend or descend three Feet, or 3 Inches, viz. from (G) to (H.) So the Tube (RM) being less than the Tube (NQ) in Proportion as (1) is less than three. The Water must descend from (G) to (H) to raise the Water in the Tube (NQ) from (O) to (P) which is ½ of the Height. And which is admirable, if the Liquor from (O) to (N) be Wine by turning the Cock (S) gently the Water shall carry up the Wine in an entire Body. And this is a pretty Experiment in hydrostatics; and these Cautionary Reflections will, I presume, if throughly understood, discourage young Practitioners from ever attempting to deceive the Order of Nature, and confound the Equilibrium of Weights (whether liquid or dry) by imaginary Perpetual Motions. THE LAST TABLE. A most Excellent TABLE of POLIGONES from 3 to 80, Calculated to a Radus of 10,000,000, by that incomparable Master of Numbers, LUDOLPHUS A CULEN, Published Anno Dom. 1619. Pol. 3 17,320,508 4 14,242,135 5 11,755,705 6 10,000,000 7 8,677,674 8 7,653,668 9 6,840,402 10 6,180,339 11 5,634,651 12 5,176,380 13 4,786,313 14 4,450,418 15 4,158,233 16 3,901,806 17 3,674,990 18 3,472,993 19 3,291,891 20 3,128,689 21 2,980,845 22 2,846,296 23 2,723,332 24 2,610,523 25 2,506,660 26 2,410,733 27 2,321,858 28 2,239,289 29 2,162,380 30 2,090,569 31 2,023,366 32 1,960,342 33 1,901,120 34 1,845,362 35 1,792,786 36 1,743,114 37 1,696,118 38 1,651,586 39 1,609,331 40 1,561,181 41 1,530,985 42 1,494,601 43 1,459,906 44 1,426,783 45 1,395,129 46 1,364,848 47 1,335,852 48 1,308,062 49 1,281,404 50 1,255,810 51 1,231,218 52 1,207,569 53 1,184,812 54 1,162,896 55 1,141,776 56 1,121,408 57 1,101,755 58 1,082,778 59 1,074,453 60 1,046,719 61 1,029,575 62 1,012,983 63 0,996,912 64 0,981,353 65 0,966,275 66 0,951,638 67 0,937,445 68 0,923,669 69 0,910,291 70 0,897,296 71 0,884,666 72 0,872,387 73 0,860,444 74 0,848,824 75 0,837,513 76 0,826,499 77 0,815,771 78 0,805,318 79 0,795,130 80 0,785,196 The Use of the TABLE of Polygones. SUppose you had a Wheel, in which you intent there should be Forty Clogs or Teeth, standing at equal Distances, and the Diameter of this Wheel be Thirty Three. By this Table you must proceed thus; First having taken half 33. which is 16.5 for the Radius of your Circle, look out the side of the Polygon of 40, standing over against that Number in the Table, which is 1,561. Then say, As 10.000. 1.561. so 16. 5. 2.575. this taken off a Line of equal parts, by which you measure your Radius, will rightly divide your Circle; and each of these Points of Division so found, will be Centres for your Teeth or Clogs. There are other Operations relating to the Division of Wheels, or Circles, which may be nicely performed by the Assistance of this Table; but the Nature of it being understood, 'tis easy to apply it to those other Uses. Note, To shorten the Work, I thought it convenient to take a lesser Radius, as 10.000. and 1.561, which stands over against 40 in the Table, will consist of the first Figure an Integer, and the rest Decimals; and as you take a greater or a less Radius, you must make use of more or fewer Figures, throughout the whole Table. FINIS. BOOKS Printed for John Laurence, at the Angel in the Poultry. MR. Pools English Annotations. Folio. The life of the Reverend Mr. Richard Baxter. Folio. Mr. Lorimer's Apology for the Ministers who subscribed only unto the Stating of the Truths and Errors in Mr. William's Book, in answer to Mr. Trail's Letter to a Minister in the Country. 4to. An Answer of Mr. Giles Firmin to Mr. Grantham, about Infant Baptism. 4o. Some Remarks upon two Anabaptist Pamphlets. By Giles Firmin. 4to. Mr. Firmin's Review of Richard Davis his Vindication. 4to. A Proposal to perform Music, in perfect and Mathematical Proportions. By Tho. Salmon, Rector of Mepsal in Bedfordshire. 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