Lines and Angles.

Lines and Angles.

Note 1. The observation will be more exact, if, at the point D, a staff be placed in the ground perpendicularly, over the top of which the observator may see a point of the glass exactly in a line betwixt him and the tower.

Note 2. In place of a mirror may be used the surface of water contained in a vessel, which naturally becomes parallel to the horizon.

PROPOSITION VI.

Fig. 7. To measure an inaccessible height AB by means of two staffs.—Let there be placed perpendicularly in the ground a longer staff DE, likewise a shorter one FG, so as the observator may see A, the top of the height to be measured, over the ends DF of the two staffs; let FH and DC, parallel to the horizon, meet DE and AB in H and C; then the triangles FHD, DCA, shall be equiangular; for the angles at C and H are right ones; likewise the angle A is equal to the angle FDH; wherefore the remaining angles DFH, and ADC, are also equal: wherefore, as FH, the distance of the staffs, to HD, the excess of the longer staff above the shorter; so is DC, the distance of the longer staff from the tower, to CA, the excess of the height of the tower above the longer staff. And thence CA will be found by the rule of three.

To which, if the length DE be added, you will have the whole height of the tower BA.

SCHOLIUM.

Fig. 8. Many other methods may be occasionally contrived for measuring an inaccessible height. For example, from the given length of the shadow BD, to find out the height AB, thus: Let there be erected a staff CE perpendicularly, producing the shadow EF; the triangles ABD, CEF, are equiangular; for the angles at B and E are right; and the angles ADB and CFE are equal, each being equal to the angle of the sun's elevation above the horizon: Therefore, as EF, the shadow of the staff, to EC, the staff itself; so BD, the shadow of the tower, to BA, the height of the tower. Though the plane on which the shadow of the tower falls be not parallel to the horizon, if the staff be erected in the same plane, the rule will be the same.

PROPOSITION VII.

To measure an inaccessible height by means of two staffs.—Hitherto we have supposed the height to be accessible, or that we can come at the lower end of it; now if, because of some impediment, we cannot get to a tower, or if the point whose height is to be found out be the summit of a hill, so that the perpendicular be hid within the hill; if, for want of better instruments, such an inaccessible height is to be measured by means of two staffs, let the first observation be made with the staffs DE and FG, (as in prop. 6.); then the observator is to go off in a direct line from the height and first station, till he come to the second station; where (fig. 11.) he is to place the longer staff perpendicularly at RN, and the shorter staff at KO, so that the summit A may be seen along their tops; that is, so that the points KNA may be in the same right line. Through the point N, let there be drawn the right line NP parallel to FA: Wherefore in the triangles KNK, KAF, the angles KNP, KAF are equal, also the angle AKF is common to both; consequently the remaining angle KPN is equal to the remaining angle KFA. And therefore, PN : FA :: KP : KF. But the triangles PNL, FAS are similar; therefore, PN : FA :: NL : SA. Therefore (by the 11. 5. Eucl.) KP : KF :: NL : SA. Thence, alternately, it will be, as KP (the excess of the greater distance of the short staff from the long one above its lesser distance from it) to NL, the excess of the longer staff above the shorter; so KF, the distance of the two stations of the shorter staff to SA, the excess of the height fought above the height of the shorter staff. Therefore SA will be found by the rule of three. To which let the height of the shorter staff be added, and the sum will give the whole inaccessible height BA.

Note 1. In the same manner may an inaccessible height be found by a geometrical square, or by a plain spectulum. But we shall leave the rules to be found out by the student, for his own exercise.

Note 2. That by the height of the staff we understand its height above the ground in which it is fixed.

Note 3. Hence depends the method of using other instruments invented by geometricians; for example, of the geometrical cross: and if all things be justly weighed, a like rule will serve for it as here. But we incline to touch only upon what is most material.

PROPOSITION VIII.

Fig. 9. To measure the distance AB, to one of whose extremities we have access, by the help of four staffs.—Let there be a staff fixed at the point A; then going back at some sensible distance in the same right line, let another be fixed in C, so that both the points A and B be covered and hid by the staff C; likewise going off in a perpendicular from the right line CB, at the point A (the method of doing which shall be shown in the following scholium), let there be placed another staff at H; and in the right line CKG (perpendicular to the same CB, at the point B), and at the point of it K, such that the points K, H, and B may be in the same right line, let there be fixed a fourth staff. Let there be drawn, or let there be supposed to be drawn, a right line GH parallel to CA. The triangles KGH, HAB, will be equiangular; for the angles HAB, KGH are right angles. Also the angles ABH, KHG are equal; wherefore, as KG (the excess of CK above AH) to GH, or to CA, the distance betwixt the first and second staff; so is AH, the distance betwixt the first and third staff, to AB the distance sought.

SCHOLIUM.

Fig. 10. To draw on a plane a right line AE perpendicular to CH, from a given point A; take the right lines AB, AD, on each side equal; and in the points B and D, let there be fixed stakes, to which let there be tied two equal ropes BE, DE, or one having a mark in the middle, and holding in your hand their extremities joined (or the mark in the middle, if it be but one), draw out the ropes on the ground; and then then where the two ropes meet, or at the mark, when by it the rope is fully stretched, let there be placed a third flake at E; the right line AE will be perpendicular to CH in the point A (prob. I. of Part I.). In a manner not unlike to this, may any problems, that are resolved by the square and compasses, be done by ropes and a cord turned round as a radius.

**PROPOSITION IX.**

Fig. 12. To measure the distance AB, one of whose extremities is accessible.—From the point A, let the right line AC of a known length be made perpendicular to AB (by the preceding scholium): likewise draw the right line CD perpendicular to CB, meeting the right line AB in D; then as DA : AC :: AC : AB. Wherefore, when DA and AC are given, AB will be found by the rule of three.

**SCHOLIUM.**

All the preceding operations depend on the equality of some angles of triangles, and on the similarity of the triangles arising from that equality. And on the same principles depend innumerable other operations which a geometrician will find out of himself, as is very obvious. However, some of these operations require such exactness in the work, and without it are so liable to errors, that, ceteris paribus, the following operations, which are performed by a trigonometrical calculation, are to be preferred; yet could we not omit those above, being most easy in practice, and most clear and evident to those who have only the first elements of geometry. But if you are provided with instruments, the following operations are more to be relied upon. We do not insist on the easiest cases to those who are skilled in plain trigonometry, which is indeed necessary to anyone who would apply himself to practice. See Trigonometry.

**PROPOSITION X.**

Fig. 13. To describe the construction and use of the geometrical quadrant.—The geometrical quadrant is the fourth part of a circle divided into 90 degrees, to which two sights are adapted, with a perpendicular or plumb-line hanging from the centre. The general use of it is for investigating angles in a vertical plane, comprehended under right lines going from the centre of the instrument, one of which is horizontal, and the other is directed to some visible point. This instrument is made of any solid matter, as wood, copper, &c.

**PROPOSITION XI.**

Fig. 14. To describe and make use of the graphometer.—The graphometer is a semicircle made of any hard matter, of wood, for example, or brass, divided into 180 degrees; so fixed on a fulcrum, by means of a brass ball and socket, that it easily turns about, and retains any situation; two sights are fixed on its diameter. At the centre there is commonly a magnetic needle in a box. There is likewise a moveable ruler, which turns round the centre, and retains any situation given it. The use of it is to observe any angle, whose vertex is at the centre of the instrument in any plane (though it is most commonly horizontal, or nearly so), and to find how many degrees it contains.

**PROPOSITION XII.**

Fig. 15. and 16. To describe the manner in which angles are measured by a quadrant or graphometer.—Let there be an angle in a vertical plane, comprehended between a line parallel to the horizon HK, and the right line RA, coming from any remarkable point of a tower or hill, or from the sun, moon, or a star. Suppose that this angle RAH is to be measured by the quadrant: let the instrument be placed in the vertical plane, so as that the centre A may be in the angular point; and let the sights be directed towards the object at R (by the help of the ray coming from it, if it be the sun or moon, or by the help of the visual ray, if it is anything else), the degrees and minutes in the arc BC, cut off by the perpendicular, will measure the angle RAH required. For, from the make of the quadrant, BAD is a right angle; therefore BAR is likewise right, being equal to it. But, because HK is horizontal, and AC perpendicular, HAC will be a right angle; and therefore equal also to BAR. From those angles subtract the part HAB that is common to both; and there will remain the angle BAC equal to the angle RAH. But the arc BC is the measure of the angle BAC; consequently, it is likewise the measure of the angle RAH.

Note, That the remaining arc on the quadrant DC is the measure of the angle RAZ, comprehended between the foresaid right line RA and AZ which points to the zenith.

Let it now be required to measure the angle ACB (fig. 16.) in any plane, comprehended between the right lines AC and BC, drawn from two points A and B, to the place of station C. Let the graphometer be placed at C, supported by its fulcrum (as was shown above); and let the immovable sights on the side of the instrument DE be directed towards the point A; and likewise (while the instrument remains immovable) let the sights of the ruler FG (which is moveable about the centre C) be directed to the point B. It is evident that the moveable ruler cuts off an arc DFI, which is the measure of the angle ACB sought. Moreover, by the same method, the inclination of CE, or of FG, may be observed with the meridian line, which is pointed out by the magnetic needle inclosed in the box, and is moveable about the centre of the instrument, and the measure of this inclination or angle found in degrees.

**PROPOSITION XIII.**

Fig. 17. To measure an inaccessible height by the geometrical quadrant.—By the 12th prop. of this Part, let the angle C be found by means of the quadrant. Then in the triangle ABC, right-angled at B (BC being supposed the horizontal distance of the observer from the tower), having the angle at C, and the side BC, the required height BA will be found by the 3d case of plane trigonometry. See Trigonometry.

**PROPOSITION XIV.**

Fig. 18. To measure an inaccessible height by the geometrical quadrant.—Let the angle ACB be observed with the quadrant (by the 12th prop. of this Part); then let the observer go from C to the second station D, in the right line BCD (provided BCD be a horizontal plane); and after measuring this distance CD, take the angle ADC likewise with the quadrant. Then, in the triangle ACD, there is given the angle ADC, with the angle ACD; because ACB was given before; therefore (by art. 59. of Part I.) the remaining angle CAD is given likewise. But the side CD is likewise Part II.

Lines and likewise given, being the distance of the station C and Angles D; therefore (by the first case of oblique-angled triangles in trigonometry) the side AC will be found. Wherefore, in the right-angled triangle ABC, all the angles and the hypotenuse AC are given; consequently, by the fourth case of trigonometry, the height sought AB will be found; as also (if you please) the distance of the station C, from AB the perpendicular within the hill or inaccessible height.

PROPOSITION XV.

Fig. 19. From the top of a given height, to measure the distance BC.—Let the angle BAC be observed by the 12th prop. of this; wherefore in the triangle ABC, right-angled at B, there is given by observation the angle at A; whence (by the 59th att. of Part I.) there will also be given the angle BCA; moreover the side AB (being the height of the tower) is supposed to be given. Wherefore, by the 3d case of trigonometry, BC, the distance sought, will be found.

PROPOSITION XVI.

Fig. 20. To measure the distance of two places A and B, of which one is accessible, by the graphometer.—Let there be erected at two points A and C, sufficiently distant, two visible signs; then (by the 12th prop. of this Part) let the two angles BAC, BCA, be taken by the graphometer. Let the distance of the stations A and C be measured with a chain. Then the third angle B being known, and the side AC being likewise known; therefore, by the first case of trigonometry, the distance required, AB, will be found.

PROPOSITION XVII.

Fig. 21. To measure by the graphometer the distance of two places, neither of which is accessible.—Let two stations C and D be chosen, from each of which the places may be seen whose distance is sought; let the angles ACD, ACB, BCD, and likewise the angles BDC, BDA, CDA, be measured by the graphometer; let the distance of the stations C and D be measured by a chain, or (if it be necessary) by the preceding practice. Now, in the triangle ACD, there are given two angles ACD and ADC; therefore, the third CAD is likewise given; moreover, the side CD is given; therefore, by the first case of trigonometry, the side AD will be found. After the same manner, in the triangle BCD, from all the angles and one side CD given, the side BD is found. Therefore, in the triangle ADB, from the given sides DA and DB, and the angle ADB contained by them, the side AB (the distance sought) is found by the 4th case of trigonometry of oblique-angled triangles.

PROPOSITION XVIII.

Fig. 22. It is required by the graphometer and quadrant to measure an accessible height AB, placed so on a slope, that one can neither go near it in an horizontal plane, nor recede from it, as we supposed in the solution of the 14th prop.—Let there be chosen any situation, as C, and another D; where let some mark be erected; let the angles ACD and ADC be found by the graphometer; then the third angle DAC will be known. Let the side CD, the distance of the stations, be measured with a chain, and thence (by trigon.) the side AC will be found. Again, in the triangle ACB, right-angled at B, having found by the quadrant the angle ACB, the other angle CAB is known likewise; but the side AC in the triangle ADC is already known; therefore the height required AB will be found by the lines and 4th case of right-angled triangles. If the height of the tower is wanted, the angle BCF will be found by the quadrant: which being taken from the angle ACB already known, the angle ACF will remain; but the angle FAC was known before; therefore the remaining angle AFC will be known. But the side AC was also known before; therefore, in the triangle AFC, all the angles and one of the sides AC being known, AF, the height of the tower above the hill, will be found by trigonometry.

SCHOLIUM.

It were easy to add many other methods of measuring heights and distances; but if what is above be understood, it will be easy (especially for one that is versed in the elements) to contrive methods for this purpose, according to the occasion: so that there is no need of adding any more of this sort. We shall subjoin here a method by which the diameter of the earth may be found out.

PROPOSITION XIX.

Fig. 1. To find the diameter of the earth from one observation.—Let there be chosen a high hill AB, near the sea-shore, and let the observer on the top of it, with an exact quadrant divided into minutes and seconds by transverse divisions, and fitted with a telescope in place of the common sights, measure the angle ABE contained under the right line AB, which goes to the centre, and the right line BE drawn to the sea, a tangent to the globe at E; let there be drawn from A perpendicular to BD, the line AF meeting BE in F. Now in the right-angled triangle BAF all the angles are given, also the side AB, the height of the hill; which is to be found by some of the foregoing methods as exactly as possible; and (by trigonometry) the sides BF and AF are found. But by cor. 36th 3. Eucl. AF is equal to FE; therefore BE will be known. Moreover, by 36th 3. Eucl. the rectangle under BA and BD is equal to the square of BE. And thence by 17th 6. Eucl., as AB : BE :: BE : BD. Therefore, since AB and BE are already given, BD will be found by 11th 6. Eucl. or by the rule of three; and subtracting BA, there will remain AD the diameter of the earth sought.

SCHOLIUM.

Many other methods might be proposed for measuring the diameter of the earth. The most exact is that proposed by Mr Picart of the academy of sciences at Paris.

"According to Mr Picart, a degree of the meridian at the latitude of 49° 21' was 57,060 French toises, each of which contains six feet of the same measure: from which it follows, that if the earth be an exact sphere, the circumference of a great circle of it will be 123,249,600 Paris feet, and the semidiameter of the earth 19,615,800 feet: but the French mathematicians, who of late have examined Mr Picart's operations, assure us, that the degree in that latitude is 57,183 toises. They measured a degree in Lapland, in the latitude of 66° 26', and found it of 57,438 toises. By comparing these degrees, as well as by the observations on pendulums, and the theory of gravity, it appears that the earth is an oblate spheroid; and (supposing those degrees to be accurately measured) the axis or diameter that passes through the poles will be the..." diameter of the equator as 177 to 178, or the earth will be 22 miles higher at the equator than at the poles.

A degree has likewise been measured at the equator, and found to be considerably less than at the latitude of Paris; which confirms the oblate figure of the earth. But an account of this last measurement has not been published as yet. If the earth was of an uniform density from the surface to the centre, then, according to the theory of gravity, the meridian would be an exact ellipsis, and the axis would be to the diameter of the equator as 230 to 231; and the difference of the semidiameter of the equator and semiaxis about 17 miles."

In what follows, a figure is often to be laid down on paper, like to another figure given; and because this likeness consists in the equality of their angles, and in the sides having the same proportion to each other (by the definitions of the 6th of Eucl.) we are now to show what methods practical geometricians use for making on paper an angle equal to a given angle, and how they constitute the sides in the same proportion. For this purpose they make use of a protractor (or, when it is wanting, a line of chords), and of a line of equal parts.

**PROPOSITION XX.**

Fig. 2, 3, 4, 5, and 6. To describe the construction and use of the protractor, of the line of chords, and of the line of equal parts. The protractor is a small semicircle of brass, or such solid matter. The semicircumference is divided into 180 degrees. The use of it is, to draw angles on any plane, as on paper, or to examine the extent of angles already laid down. For this last purpose, let the small point in the centre of the protractor be placed above the angular point, and let the side AB coincide with one of the sides that contain the angle proposed; the number of degrees cut off by the other side, computing on the protractor from B, will show the quantity of the angle that is to be measured.

But if an angle is to be made of a given quantity on a given line, and at a given point of that line, let AB coincide with the given line, and let the centre A of the instrument be applied to that point. Then let there be a mark made at the given number of degrees; and a right line drawn from that mark to the given point, will constitute an angle with the given right line of the quantity required; as is manifest.

This is the most natural and easy method, either for examining the extent of an angle on paper, or for describing on paper an angle of a given quantity.

But when there is scarcity of instruments, or because a line of chords is more easily carried about (being described on a ruler on which there are many other lines besides), practical geometricians frequently make use of it. It is made thus: let the quadrant of a circle be divided into 90 degrees (as in fig. 4.) The line AB is the chord of 90 degrees; the chord of every arc of the quadrant is transferred to this line AB, which is always marked with the number of degrees in the corresponding arc.

Note. That the chord of 60 degrees is equal to the radius, by corol. 15. 4th Eucl. If now a given angle EDF is to be measured by the line of chords from the centre D, with the distance DG (the chord of 60 degrees), describe the arc GF; and let the points G and F be marked where this arc intersects the sides of the angle. Then if the distance GF, applied on the line of chords from A to B, gives (for example) 25 degrees, this shall be the measure of the angle proposed.

When an obtuse angle is to be measured with this line, let its complement to a semicircle be measured, and thence it will be known. It were easy to transfer to the diameter of a circle the chords of all arches to the extent of a semicircle; but such are rarely found marked upon rules.

But now, if an angle of a given quantity, suppose of 50 degrees, is to be made at a given point M of the right line KL (fig. 6.) From the centre M, and the distance MN, equal to the chord of 60 degrees, describe the arc QN. Take off an arc NR, whose chord is equal to that of 50 degrees on the line of chords; join the points M and R; and it is plain that MR shall contain an angle of 50 degrees with the line KL proposed.

But sometimes we cannot produce the sides till they be of the length of a chord of 60 degrees on our scale; in which case it is fit to work by a circle of proportions (that is a sector), by which an arc may be made of a given number of degrees to any radius.

The quantities of angles are likewise determined by other lines usually marked upon rules, as the lines of fines, tangents, and secants; but as these methods are not so easy or so proper in this place, we omit them.

To delineate figures similar or like to others given, besides the equality of the angles, the same proportion is to be preserved among the sides of the figure that is to be delineated, as is among the sides of the figures given. For which purpose, on the rules used by artists, there is a line divided into equal parts, more or less in number, and greater or less in quantity according to the pleasure of the maker.

A foot is divided into inches; and an inch, by means of transverse lines, into 100 equal parts; so that with this scale, any number of inches below 12, with any part of an inch, can be taken by the compasses, providing such part be greater than the 100th part of an inch. And this exactness is very necessary in delineating the plans of houses, and in other cases.

**PROPOSITION XXI.**

Fig. 7. To lay down on paper, by the protractor or line of chords, and line of equal parts, a right-lined figure like to one given, providing the angles and sides of the figure given be known by observation or mensuration. For example, suppose that it is known that in a quadrangular figure, one side is of 235 feet, that the angle contained by it and the second side is of 84°, the second side of 288 feet, the angle contained by it and the third side of 72°, and that the third side is 294 feet. These things being given, a figure is to be drawn on paper like to this quadrangular figure. On your paper at a proper point A, let a right line be drawn, upon which take 235 equal parts, as AB. The part representing a foot is taken greater or less, according as you would have your figure greater or less. In the adjoining figure, the 100th part of an inch is taken for a foot. And accordingly an inch divided into 100 parts, and annexed to the figure, is called a scale of 100 feet. Let there be made at the point B (by the preceding proposition) an angle ABC of 85°, and let BC be taken of 288 parts like to the former. Then let the angle BCD be made of 72°, and the side CD of Part II.

Geometry.

Lines and Angles.

Then let the side AD be drawn; and it will complete the figure like to the given. The measures of the angle A and D can be known by the protractor or line of chords, and the side AD by the line of equal parts; which will exactly answer to the corresponding angles and to the side of the primary figure.

After the very same manner, from the sides and angles given which bound any right-lined figure, a figure like to it may be drawn, and the rest of its sides and angles be known.

Corollary.

Hence any trigonometrical problem in right-lined triangles may be resolved by delineating the triangle from what is given concerning it, as in this proposition. The unknown sides are examined by a line of equal parts, and the angles by a protractor or line of chords.

Proposition XXII.

The diameter of a circle being given, to find its circumference nearly.—The periphery of any polygon inscribed in the circle is less than the circumference, and the periphery of any polygon described about a circle is greater than the circumference. Whence Archimedes first discovered that the diameter was in proportion to the circumference, as 7 to 22 nearly; which serves for common use. But the moderns have computed the proportion of the diameter to the circumference to greater exactness. Supposing the diameter 100, the periphery will be more than 314, but less than 315. The diameter is more nearly to the circumference, as 113 to 355. But Ludolphus van Cuelen exceeded the labours of all; for by immense study he found, that supposing the diameter 100,000,000,000,000,000,000,000,000,000,000, the periphery will be less than 314,159,265,358,979,323,846,264,338,327,951, but greater than 314,159,265,358,979,323,846,264,338,327,950; whence it will be easy, any part of the circumference being given in degrees and minutes, to assign it in parts of the diameter.

Chap. II. Of Surveying and Measuring of Land.

Hitherto we have treated of the measuring of angles and sides, whence it is abundantly easy to lay down a field, a plane, or an entire country; for to this nothing is requisite but the proportion of triangles, and of other plain figures, after having measured their sides and angles. But as this is esteemed an important part of practical geometry, we shall subjoin here an account of it with all possible brevity; suggesting withal, that a surveyor will improve himself more by one day's practice than by a great deal of reading.

Proposition XXIII.

To explain what surveying is, and what instruments Surveyors use.—First, it is necessary that the surveyor view the field that is to be measured, and investigate its sides and angles, by means of an iron chain (having a particular mark at each foot of length, or at any number of feet, as may be most convenient for reducing lines or surfaces to the received measures), and the graphometer described above. Secondly, It is necessary to delineate the field in plan, or to form a map of it; that is, to lay down on paper a figure similar to the field; which is done by the protractor (or line of chords) and the line of equal parts. Thirdly, It is necessary to find out the area of the field so surveyed and represented by a map. Of this last we are to treat below.

The sides and angles of small fields are surveyed by the help of a plain-table: which is generally of an oblong rectangular figure, and supported by a fulcrum, so as to turn every way by means of a ball and socket. It has a moveable frame, which surrounds the board, and serves to keep a clean paper put on the board close and right to it. The sides of the frame facing the paper are divided into equal parts every way. The board hath besides a box with a magnetic needle, and moreover a large index with two sights. On the edge of the frame of the board are marked degrees and minutes, so as to supply the room of a graphometer.

Proposition XXIV.

Fig. 8. To delineate a field by the help of a plain-table, from one station whence all its angles may be seen and their distances measured by a chain.—Let the field that is to be laid down be ABCDE. At any convenient place F, let the plain-table be erected; cover it with clean paper, in which let some point near the middle represent the station. Then applying at this place the index with the sights, direct it so as that through the sights some mark may be seen at one of the angles, suppose A; and from the point F, representing the station, draw a faint right line along the side of the index: then, by the help of the chain, let FA the distance of the station from the foreaid angle be measured. Then taking what part you think convenient for a foot or pace from the line of equal parts, set off on the faint line the parts corresponding to the line FA that was measured; and let there be a mark made representing the angle of the field A. Keeping the table immovable, the same is to be done with the rest of the angles; then right lines joining those marks shall include a figure like to the field, as is evident from 5, 6. Eucl.

Corollary.

The same thing is done in like manner by the graphometer: for having observed in each of the triangles, AFB, BFC, CFD, &c. the angle at the station F, and having measured the lines from the station to the angles of the field, let similar triangles be projected on paper (by the 21. prop. of this), having their common vertex in the point of station. All the lines, excepting those which represent the sides of the field, are to be drawn faint or obscure.

Note 1. When a surveyor wants to lay down a field, let him place distinctly in a register all the observations of the angles, and the measures of the sides, until, at time and place convenient, he draw out the figure on paper.

Note 2. The observations made by the help of the graphometer are to be examined: for all the angles about the point F ought to be equal to four right ones. (by cor. 2. art. 30. of Part I.)

Proposition XXV.

Fig. 9. To lay down a field by means of two stations, from each of which all the angles can be seen, by measuring only the distance of the stations.—Let the instrument be placed at the station F; and having chosen a point representing it upon the paper which is laid upon the plain table, let the index be applied at this point, so as to be moveable about it. Then let it be directed successively to the several angles of the field; and when any angle is seen through the sights, draw an obscure line along the side of the index. Let the index, with the sights, be directed after the same manner to the station G: on the obscure line drawn along its side, pointing to A, set off from the scale of equal parts a line corresponding to the measured distance of the stations, and this will determine the point G. Then remove the instrument to the station G, and applying the index to the line representing the distance of the stations, place the instrument so that the first station may be seen through the sights. Then the instrument remaining immoveable, let the index be applied to the point representing the second station G, and be successively directed by means of its sights to all the angles of the field, drawing (as before) obscure lines: and the intersection of the two obscure lines that were drawn to the same angle from the two stations will always represent that angle on the plan. Care must be taken that those lines be not mistaken for one another. Lines joining those intersections will form a figure on the paper like to the field.

SCHOLIUM.

It will not be difficult to do the same by the graphometer, if you keep a distinct account of your observations of the angles made by the line joining the stations, and the lines drawn from the stations to the respective angles of the field. And this is the most common manner of laying down whole countries. The tops of two mountains are taken for two stations, and their distance is either measured by some of the methods mentioned above, or is taken according to common report. The sights are successively directed towards cities, churches, villages, forts, lakes, turnings of rivers, woods, &c.

Note, The distance of the stations ought to be great enough, with respect to the field that is to be measured; such ought to be chosen as are not in a line with any angle of the field. And care ought to be taken likewise that the angles, for example, FAG, FDG, &c., be neither very acute, nor very obtuse. Such angles are to be avoided as much as possible; and this admonition is found very useful in practice.

PROPOSITION XXVI.

Fig. 10. To lay down any field, however irregular its figure may be, by the help of the graphometer.—Let ABCDEFG be such a field. Let its angles (in going round it) be observed with a graphometer (by the 12th of this) and noted down; let its sides be measured with a chain; and (by what was said on the 21st of this) let a figure like to the given field be protracted on paper. If any mountain is in the circumference, the horizontal line hid under it is to be taken for a side, which may be found by two or three observations according to some of the methods described above; and its place on the map is to be distinguished by a shade, that it may be known a mountain is there.

If not only the circumference of the field is to be laid down on the plan, but also its contents, as villages, gardens, churches, public roads, we must proceed in this manner.

Let there be (for example) a church F, to be laid down in the plan. Let the angles ABF BAF be observed and protracted on paper in their proper places, the intersection of the two sides BF and AF will give the place of the church on the paper; or, more exactly, the lines BF AF being measured, let circles be described from the centres B and A, with parts from the scale corresponding to the distances BF and AF, and the place of the church will be at their intersection.

Note 1. While the angles observed by the graphometer are taken down, you must be careful to distinguish the external angles, as E and G, that they may be rightly protracted afterwards on paper.

Note 2. Our observations of the angles may be examined by computing if all the internal angles make twice as many right angles, four excepted, as there are sides of the figure; (for this is demonstrated by 32, 1. Eucl.) But in place of any external angle DEC, its complement to a circle is to be taken.

PROPOSITION XXVII.

Fig. 11. To lay down a plain field without instruments.—If a small field is to be measured, and a map of it to be made, and you are not provided with instruments; let it be supposed to be divided into triangles, by right-lines, as in the figure; and after measuring the three sides of any of the triangles, for example of ABC, let its sides be laid down from a convenient scale on paper, (by the 22nd of this.) Again, let the other two sides BD CD of the triangle CBD be measured and protracted on the paper by the same scale as before. In the same manner proceed with the rest of the triangles of which the field is composed, and the map of the field will be perfected; for the three sides of a triangle determine the triangle; whence each triangle on the paper is similar to its correspondent triangle in the field, and is similarly situated; consequently the whole figure is like to the whole field.

SCHOLIUM.

If the field be small, and all its angles may be seen from one station, it may be very well laid down by the plain-table, (by the 24th of this.) If the field be larger, and have the requisite conditions, and great exactness is not expected, it likewise may be plotted by means of the plain-table, or by the graphometer (according to the 25th of this); but in fields that are irregular and mountainous, when an exact map is required, we are to make use of the graphometer (as in the 26th of this), but rarely of the plain-table.

Having protracted the bounding lines, the particular parts contained within them may be laid down by the proper operations for this purpose (delivered in the 26th proposition; and the method described in the 27th proposition may be sometimes of service); for we may trust more to the measuring of sides than to the observing of angles. We are not to compute four-sided and many-sided figures till they are resolved into triangles: for the sides do not determine those figures.

In the laying down of cities, or the like, we may make use of any of the methods described above that may be most convenient.

The map being finished, it is transferred on clean paper, by putting the first sketch above it, and marking the angles by the point of a small needle. These points being joined by right lines, and the whole illuminated... minated by colours proper to each part, and the figure of the mariner's compass being added to distinguish the north and south, with a scale on the margin, the map or plan will be finished and neat.

We have thus briefly and plainly treated of surveying, and shown by what instruments it is performed; having avoided those methods which depend on the magnetic needle, not only because its direction may vary in different places of a field (the contrary of this at least doth not appear,) but because the quantity of an angle observed by it cannot be exactly known; for an error of two or three degrees can scarcely be avoided in taking angles by it.

As for the remaining part of surveying, whereby the area of a field already laid down on paper is found in acres, roods; or any other superficial measures; this we leave to the following section, which treats of the mensuration of surfaces.

"Besides the instruments described above, a surveyor ought to be provided with an off-set staff equal in length to 10 links of the chain, and divided into 10 equal parts. He ought likewise to have 10 arrows or small straight sticks near two feet long, fitted with iron ferrils. When the chain is first opened, it ought to be examined by the off-set staff. In measuring any line, the leader of the chain is to have the 10 arrows at first setting out. When the chain is stretched in the line, and the near end touches the place from which you measure, the leader sticks one of the 10 arrows in the ground, at the far end of the chain. Then the leader leaving the arrow, proceeds with the chain another length; and the chain being stretched in the line, so that the near end touches the first arrow, the leader sticks down another arrow at his end of the chain. The line is preserved straight, if the arrows be always set so as to be in a right line with the place you measure from, and that to which you are going. In this manner they proceed till the leader have no more arrows. At the eleventh chain, the arrows are to be carried to him again, and he is to stick one of them into the ground, at the end of the chain. And the same is to be done at the 21, 31, 41, &c., chains, if there are so many in a right line to be measured. In this manner you can hardly commit an error in numbering the chains, unless of 10 chains at once.

The off-set staff serves for measuring readily the distances of any things proper to be represented in your plan, from the station-line, while you go along. These distances ought to be entered into your field-book, with the corresponding distances from the last station, and proper remarks, that you may be enabled to plot them justly, and be in no danger of mistaking one for another when you extend your plan. The field-book may be conveniently divided into five columns. In the middle column the angles at the several stations taken by the theodolite are to be entered, with the distances from the stations. The distances taken by the off-set staff, on either side of the station-line, are to be entered into columns on either side of the middle column, according to their position with respect to that line. The names and characters of the objects, with proper remarks, may be entered in columns on either side of these last.

"Because, in the place of the graphometer described by our author, surveyors now make use of the theodolite, we shall subjoin a description of Mr Sisson's latest improved theodolite from Mr Gardner's practical surveying improved. See a figure of it in Plate CCXVIII.

"In this instrument, the three staffs, by brass ferrils, at top screw into bell-metal joints, that are moveable between brass pillars, fixed in a strong brass plate; in which, round the centre, is fixed a socket with a ball moveable in it, and upon which the four screws press, that set the limb horizontal: Next above is another such plate, through which the said screws pass, and on which, round the centre, is fixed a frustum of a cone of bell-metal, whose axis (being connected with the centre of the bell) is always perpendicular to the limb, by means of a conical brass ferril fitted to it, whereon is fixed the compass-box; and on it the limb, which is a strong bell-metal ring, whereon are moveable three brass indexes, in whose plate are fixed four brass pillars, that, joining at top, hold the centre pin of the bell-metal double sextant, whose double index is fixed on the centre of the same plate: Within the double sextant is fixed the spirit level, and over it the telescope.

"The compass-box is graved with two diamonds for north and south, and with 20 degrees on both sides of each, that the needle may be set to the variation, and its error also known.

"The limb has two fleurs de luce against the diamonds in the box, instead of 180 each, and is curiously divided into whole degrees, and numbered to the left hand at every 10 to twice 180, having three indexes distant 120. (with Nonius's divisions on each for the decimals of a degree), that are moved by a pinion fixed below one of them, without moving the limb; and in another is a screw and spring under, to fix it to any part of the limb. It has also divisions numbered, for taking the quarter girt in inches of round timber at the middle height, when standing 10 feet horizontally distant from its centre; which at 20 must be doubled, and at 30 tripled; to which a shorter index is used, having Nonius's divisions for the decimals of an inch; but an abatement must be made for the bark, if not taken off.

"The double sextant is divided on one side from under its centre (when the spirit-tube and telescope are level) to above 60 degrees each way, and numbered at 10, 20, &c. and the double index (through which it is moveable) shows on the same side the degree and decimal of any altitude or depression to that extent by Nonius's divisions: On the other side are divisions numbered, for taking the upright height of timber, &c. in feet, when distant 10 feet; which at 20 must be doubled, and at 30 tripled; and also the quantities for reducing hypotenusal lines to horizontal. It is moveable by a pinion fixed in the double index.

"The telescope is a little shorter than the diameter of the limb, that a fall may not hurt it; yet it will magnify as much, and show a distant object as perfect as most of triple its length. In its focus are very fine cross wires, whose intersection is in the plane of the double sextant; and this was a whole circle, and turned in a lathe to a true plane, and is fixed at right angles to the limb; so that, whenever the limb is set horizontal (which is readily done by making the spi- Surveying instruments level over two screws, and the like over the other two), the double sextant and telescope are moveable in a vertical plane; and then every angle taken on the limb (though the telescope be never so much elevated or depressed) will be an angle in the plane of the horizon. And this is absolutely necessary in plotting a horizontal plane.

"If the lands to be plotted are hilly, and not in any one plane, the lines measured cannot be truly laid down on paper, without being reduced to one plane, which must be the horizontal, because angles are taken in that plane.

"In viewing your objects, if they have much altitude or depression, either write down the degree and decimal shown on the double sextant, or the links shown on the back side; which last subtracted from every chain in the station-line, leaves the length in the horizontal plane. But if the degree is taken, the following table will show the quantity.

A Table of the links to be subtracted out of every chain in hypothenusal lines of several degrees altitude, or depression, for reducing them to horizontal.

| Degrees | Links | |---------|-------| | 4° 05' | 14.07 | | 5° 73' | 16.26 | | 7° 02' | 18.195 | | 8° 11' | 19.95 | | 11° 48' | 21.565 |

"Let the first station line really measure 1107 links, and the angle of altitude or depression be 19° 95'; looking in the table you will find against 19° 95', is 6 links. Now 6 times 1107 is 66, which subtracted from 1107, leaves 1041, the true length to be laid down in the plan.

"It is useful in surveying, to take the angles, which the bounding lines form, with the magnetic needle, in order to check the angles of the figure, and to plot them conveniently afterwards."

CHAP. III. Of the Surfaces of Bodies.

The smallest superficial measure with us is a square inch; 144 of which make a square foot. Wrights make use of these in measuring deals and planks; but the square foot which the glaziers use in measuring of glass, consists only of 64 square inches. The other measures are, first, the ell square; secondly, the fall, containing 36 square ells; thirdly, the rood, containing 40 falls; fourthly, the acre, containing 4 roods. Slaters, masons, and pavers, use the ell square and the fall; surveyors of land use the square ell, the fall, the rood, and the acre.

The superficial measures of the English acre, first, the square foot; secondly, the square yard, containing 9 square feet, for their yard contains only 3 feet; thirdly, the pole, containing 30¼ square yards; fourthly, the rood, containing 40 poles; fifthly, the acre, containing 4 roods. And hence it is easy to reduce our superficial measures to the English, or theirs to ours.

"In order to find the content of a field, it is most convenient to measure the lines by the chains described above, p. 671. that of 22 yards for computing the English acres, and that of 24 Scots ells for the acres of Scotland. The chain is divided into 100 links, and the square of the chain is 10,000 square links; 10 surfaces of squares of the chain, or 100,000 square links, give an acre. Therefore, if the area be expressed by square links, divide by 100,000, or cut off five decimal places, and the quotient shall give the area in acres and decimals of an acre. Write the entire acres apart; but multiply the decimals of an acre by 4, and the product shall give the remainder of the area in roods and decimals of a rood. Let the entire roods be noted apart after the acres; then multiply the decimals of a rood by 40, and the product shall give the remainder of the area in falls or poles. Let the entire falls or poles be then writ after the roods, and multiply the decimals of a fall by 36, if the area is required in the measures of Scotland; but multiply the decimals of a pole by 30¼, if the area is required in the measures of England, and the product shall give the remainder of the area in square ells in the former case, but in square yards in the latter. If, in the former case, you would reduce the decimals of the square ell to square feet, multiply them by 9.50694; but, in the latter case, the decimals of the English square yard are reduced to square feet, by multiplying them by 9.

"Suppose, for example, that the area appears to contain 12,658.42 square links of the chain of 24 ells; and that this area is to be expressed in acres, roods, falls, &c. of the measures of Scotland. Divide the square links by 100,000, and the quotient 12.65842 shows the area to contain 12 acres 12.65842 of an acre. Multiply the decimal part by 4, and the product 2.63568 gives the remainder in roods and decimals of a rood. Those decimals of the rood being multiplied by 40, the product gives 25.3472 falls. Multiply the decimals of the fall by 36, and the product gives 12.4992 square ells. The decimals of the square ell multiplied by 9.50994 give 4.7458 square feet. Therefore the area proposed amounts to 12 acres, 2 roods, 25 falls, 12 square ells, and 4.7458 square feet.

"But if the area contains the same number of square links of Gunter's chain, and is to be expressed by English measures, the acres and roods are computed in the same manner as in the former case. The poles are computed as the falls. But the decimals of the pole, viz. 30¼, are to be multiplied by 30¼ (or 30¼), and the product gives 10.5028 square yards. The decimals of the square yard, multiplied by 9, give 4.5252 square feet; therefore, in this case, the area is in English measure 12 acres, 2 roods, 25 poles, 10 square yards, and 4.5252 square feet.

"The Scots acre is to the English acre, by statute, as 100,000 to 78,694, if we have regard to the difference betwixt the Scots and English foot above mentioned. But it is customary in some parts of England to have 18.21, &c. feet to a pole, and 160 such poles to an acre; whereas, by the statute, 16¼ feet make a pole. In such cases the acre is greater in the duplicate ratio of the number of feet to a pole.

"They who measure land in Scotland by an ell of 37 English inches, make the acre less than the true Scots acre by 592.9 square English feet, or by about ⅓ of the acre.

"An husband land contains 6 acres of stock and scythe-land, that is, of land that may be tilled with a plough, and mown with a scythe; 13 acres of arable land..." Surfaces of land make an oxgang or oxengate; four oxengate make a pound land of old extent (by a decree of the Exchequer, March 11, 1585), and is called libra terra. A forty-shilling land of old extent contains 8 oxgang, or 104 acres.

"The arpent, about Paris, contains 32,400 square Paris feet, and is equal to 2½ Scots roods, or 37½ English roods.

"The actus quadratus, according to Varro, Columella, &c., was a square of 120 Roman feet. The jugerum was the double of this. It is to the Scots acre as 10,000 to 20,456, and to the English acre as 10,000 to 16,097. It was divided (like the as) into 12 unciae, and the uncia into 24 scrupula."—This, with the three preceding paragraphs, are taken from an ingenious manuscript, written by Sir Robert Stewart professor of natural philosophy. The greatest part of the table in p. 671, was taken from it likewise.

**PROPOSITION XXVIII.**

Fig. 12. To find out the area of a rectangular parallelogram ABCD.—Let the side AB, for example, be 5 feet long, and BC (which constitutes with BA a right angle at B) be 17 feet. Let 17 be multiplied by 5, and the product 85 will be the number of square feet in the area of the figure ABCD. But if the parallelogram proposed is not rectangular as BEFC, its base BC multiplied into its perpendicular height AB (not into its side BE) will give its area. This is evident from art. 68. of Part I.

**PROPOSITION XXIX.**

Fig. 13. To find the area of a given triangle.—Let the triangle BAC be given, whose base BC is supposed 9 feet long; let the perpendicular AD be drawn from the angle A opposite to the base, and let us suppose AD to be 4 feet. Let the half of the perpendicular be multiplied into the base, or the half of the base into the perpendicular, or take the half of the product of the whole base into the perpendicular, the product gives 18 square feet for the area of the given triangle.

But if only the sides are given, the perpendicular is found either by protracting the triangle, or by 12th and 13th 2. Eucl. or by trigonometry. But how the area of a triangle may be found from the given sides only, shall be shown in the 31st proposition.

**PROPOSITION XXX.**

Fig. 14. To find the area of any rectilineal figure.—If the figure be irregular, let it be resolved into triangles; and drawing perpendiculars to the bases in each of them, let the area of each triangle be found by the preceding proposition, and the sum of these areas will give the area of the figure.

**SCHOLIUM I.**

In measuring boards, planks, and glass, their sides are to be measured by a foot-rule divided into 100 equal parts; and after multiplying the sides, the decimal fractions are easily reduced to lesser denominations. The mensuration of these is easy, when they are rectangular parallelograms.

**SCHOLIUM II.**

If a field is to be measured, let it first be plotted on paper, by some of the methods above described, and let the figure so laid down be divided into triangles, as was shown in the preceding proposition.

The base of any triangle, or the perpendicular upon the base, or the distance of any two points of the surfaces of field, is measured by applying it to the scale according to which the map is drawn.

**SCHOLIUM III.**

But if the field given be not in an horizontal plane, but uneven and mountainous, the scale gives the horizontal line between any two points, but not their distance measured on the uneven surface of the field. And indeed it would appear, that the horizontal plane is to be accounted the area of an uneven and hilly country. For if such ground is laid out for building on, or for planting with trees, or bearing corn, since these stand perpendicular to the horizon, it is plain, that a mountainous country cannot be considered as of greater extent for those uses than the horizontal plane; nay, perhaps, for nourishing of plants, the horizontal plane may be preferable.

If, however, the area of a figure, as it lies regularly on the surface of the earth, is to be measured, this may be easily done by resolving it into triangles as it lies. The sum of their areas will be the area sought; which exceeds the area of the horizontal figure more or less, according as the field is more or less uneven.

**PROPOSITION XXXI.**

Fig. 13. The sides of a triangle being given, to find the area, without finding the perpendicular.—Let all the sides of the triangle be collected into one sum; from the half of which let the sides be separately subtracted, that three differences may be found betwixt the foresaid half sum and each side; then let these three differences and the half sum be multiplied into one another, and the square root of the product will give the area of the triangle. For example, let the sides be 10, 17, 21; the half of their sum is 24; the three differences betwixt this half sum and the three sides, are 14, 7, and 3. The first being multiplied by the second, and their product by the third, we have 294 for the product of the differences; which multiplied by the foresaid half sum 24, gives 7056; the square root of which 84 is the area of the triangle. The demonstration of this, for the sake of brevity, we omit. It is to be found in several treatises, particularly in Clavius's Practical Geometry.

**PROPOSITION XXXII.**

Fig. 15. The area of the ordinate figure ABEFGH is equal to the product of the half circumference of the polygon, multiplied into the perpendicular drawn from the centre of the circumscribed circle to the side of the polygon.—For the ordinate figure can be resolved into as many equal triangles as there are sides of the figure; and since each triangle is equal to the product of half the base into the perpendicular, it is evident that the sum of all the triangles together, that is the polygon, is equal to the product of half the sum of the bases (that is the half of the circumference of the polygon) into the common perpendicular height of the triangles drawn from the centre C to one of the sides; for example, to AB.

**PROPOSITION XXXIII.**

Fig. 16. The area of a circle is found by multiplying the half of the periphery into the radius, or the half of the radius into the periphery.—For a circle is not different from an ordinate or regular polygon of an infinite number of sides, and the common height of the triangles... to which the polygon or circle may be supposed to be divided is the radius of the circle.

Were it worth while, it were easy to demonstrate accurately this proposition, by means of the inscribed and circumscribed figures, as is done in the 5th prop. of the treatise of Archimedes concerning the dimensions of the circle.

**COROLLARY.**

Hence also it appears, that the area of the sector ABCD is produced by multiplying the half of the arc into the radius, and likewise that the area of the segment of the circle ADC is found by subtracting from the area of the sector the area of the triangle ABC.

**PROPOSITION XXXIV.**

Fig. 17. The circle is to the square of the diameter as 11 to 14 nearly.—For if the diameter AB be supposed to be 7, the circumference AHBK will be almost 22 (by the 22d prop. of this Part), and the area of the square DC will be 49; and, by the preceding prop. the area of the circle will be 28½; therefore the square DC will be to the inscribed circle as 49 to 38½, or as 98 to 77, that is, as 14 to 11. Q.E.D.

If greater exactness is required, you may proceed to any degree of accuracy: for the square DC is to the inscribed circle, as 1 to 1—½+¼—½+¼—½+¼—½+¼, &c., in infinitum.

"This series will be of no service for computing the area of the circle accurately, without some further artifice, because it converges too slow a rate. The area of the circle will be found exactly enough for most purposes, by multiplying the square of the diameter by 7854, and dividing by 10,000, or cutting off four decimal places from the product; for the area of the circle is to the circumscribed square nearly as 7854 to 10,000."

**PROPOSITION XXXV.**

Fig. 18. To find the area of a given ellipse.—Let ABCD be an ellipse, whose greater diameter is BD, and the lesser AC, bisecting the greater perpendicularly in E. Let a mean proportional HF be found (by 12th 6. Eucl.) between AC and BD, and (by the 33d of this) find the area of the circle described on the diameter HF. This area is equal to the area of the ellipse ABCD. For because, as BD to AC, so the square of BD to the square of HF, (by 2. cor. 20th 6. Eucl.); but (by the 2d 12. Eucl.) as the square of BD to the square of HF, so is the circle of the diameter BD to the circle of the diameter HF; therefore as BD to AC, so is the circle of the diameter BD to the circle of the diameter HF. And (by the 5th prop. of Archimedes of spheroids) as the greater diameter BD to the lesser AC, so is the circle of the diameter BD to the ellipse ABCD. Consequently (by the 11th 5. Eucl.) the circle of the diameter BD will have the same proportion to the circle of the diameter HF, and to the ellipse ABCD. Therefore (by 9th 5. Eucl.) the area of the circle of the diameter HF will be equal to the area of the ellipse ABCD. Q.E.D.

S.C.H.O.L.I.U.M.

From this and the two preceding propositions, a method is derived of finding the area of an ellipse. There are two ways: 1st, Say, as one is to the lesser diameter, so is the greater diameter to a fourth number, (which is found by the rule of three). Then again say, as 14 to 11, so is the fourth number found to the area sought. But the second way is shorter. Multiply the lesser diameter into the greater, and the product by 11; then divide the whole product by 14, and the quotient will be the area sought of the ellipse. For example, Let the greater diameter be 30, and the lesser 7; by multiplying 10 by 7, the product is 70; and multiplying that by 11, it is 770; and dividing 770 by 14, the quotient will be 55, which is the area of the ellipse sought.

"The area of the ellipse will be found more accurately, by multiplying the product of the two diameters by 7854."

We shall add no more about other plain surfaces, whether rectilinear or curvilinear, which seldom occur in practice; but shall subjoin some propositions about measuring the surface of solids.

**PROPOSITION XXXVI.**

To measure the surface of any prism.—By the 14th definition of the 11th Eucl. a prism is contained by planes, of which two opposite sides (commonly called the bases) are plain rectilineal figures; which are either regular and ordinate, and measured by prop. 32. of this; or however irregular, and then they are measured by the 28th prop. The other sides are parallelograms, which are measured by prop. 28th; and the whole superficies of the prism consists of the sum of those taken altogether.

**PROPOSITION XXXVII.**

To measure the superficies of any pyramid.—Since its basis is a rectilineal figure, and the rest of the planes terminating in the top of the pyramid are triangles; these measured separately, and added together, give the surface of the pyramid required.

**PROPOSITION XXXVIII.**

To measure the superficies of any regular body.—These bodies are called regular, which are bounded by equilateral and equiangular figures. The superficies of the tetraedron consists of four equal and equiangular triangles; the superficies of the hexaedron or cube, of six equal squares; an octaedron, of eight equal equilateral triangles; a dodecaedron, of twelve equal and ordinate pentagons; and the superficies of an icofaedron, of twenty equal and equilateral triangles. Therefore it will be easy to measure these surfaces from what has been already shown.

In the same manner we may measure the superficies of a solid contained by any planes.

**PROPOSITION XXXIX.**

Fig. 19. To measure the superficies of a cylinder.—Because a cylinder differs very little from a prism, whose opposite planes or bases are ordinate figures of an infinite number of sides, it appears that the superficies of a cylinder, without the basis, is equal to an infinite number of parallelograms; the common altitude of all which is the same with the height of the cylinder, and the bases of them all differ very little from the periphery of the circle which is the base of the cylinder. Therefore this periphery multiplied into the common height, gives the superficies of the cylinder, excluding the bases; which are to be measured separately by the 33d proposition.

This proposition concerning the measure of the surface of the cylinder (excluding its basis) is evident from this, that when it is conceived to be spread out, it Surfaces of it becomes a parallelogram, whose base is the periphery of the circle of the base of the cylinder stretched into a right line, and whose height is the same with the height of the cylinder.

**PROPOSITION XL.**

Fig. 20. To measure the surface of a right cone.—The surface of a right cone is very little different from the surface of a right pyramid, having an ordinate polygon for its base of an infinite number of sides; the surface of which (excluding the base) is equal to the sum of the triangles. The sum of the bases of these triangles is equal to the periphery of the circle of the base, and the common height of the triangles is the side of the cone \(AB\); wherefore the sum of these triangles is equal to the product of the sum of the bases (i.e., the periphery of the base of the cone) multiplied into the half of the common height, or it is equal to the product of the periphery of the base.

If the area of the base is likewise wanted, it is to be found separately by the 33rd prop. If the surface of a cone is supposed to be spread out on a plane, it will become a sector of a circle, whose radius is the side of the cone; and the arc terminating the sector is made from the periphery of the base. Whence, by corol. 33d prop. of this, its dimension may be found.

**COROLLARY.**

Hence it will be easy to measure the surface of a frustum of a cone cut by a plane parallel to the base.

**PROPOSITION XLI.**

Fig. 21. To measure the surface of a given sphere.—Let there be a sphere, whose centre is \(A\), and let the area of its convex surface be required. Archimedes demonstrates (37th prop. 1. book of the sphere and cylinder) that its surface is equal to the area of four great circles of the sphere; that is, let the area of the great circle be multiplied by 4, and the product will give the area of the sphere; or (by the 20th 6. and 2nd 12. of Eucl.) the area of the sphere given is equal to the area of a circle whose radius is the right line \(BC\), the diameter of the sphere. Therefore having measured (by 33d prop.) the circle described with the radius \(BC\), this will give the surface of the sphere.

**PROPOSITION XLII.**

Fig. 22. To measure the surface of a segment of a sphere.—Let there be a segment cut off by the plane \(ED\). Archimedes demonstrates (49. and 50. 1. De Solidis) that the surface of this segment, excluding the circular base, is equal to the area of a circle whose radius is the right line \(BE\) drawn from the vertex \(B\) of the segment to the periphery of the circle \(DE\). Therefore (by the 33d prop.) it is easily measured.

**COROLLARY 1.**

Hence that part of the surface of a sphere that lies between two parallel planes is easily measured, by subtracting the surface of the lesser segment from the surface of the greater segment.

**COROLLARY 2.**

Hence likewise it follows, that the surface of a cylinder, described about a sphere (excluding the basis) is equal to the surface of the sphere, and the parts of the one to the parts of the other, intercepted between planes parallel to the basis of the cylinder.

---

**CHAP. IV. Of solid Figures and their Mensuration, comprehending likewise the Principles of Gauging Vessels of all Figures.**

As in the former part of this treatise we took an inch for the smallest measure in length, and an inch square for the smallest superficial measure; so now, in treating of the mensuration of solids, we take a cubical inch for the smallest solid measure. Of these, 109 make a Scots pint; other liquid measures depend on this, as is generally known.

In dry measures, the firkin, by statute, contains 19½ pints; and on this depend the other dry measures; therefore, if the content of any solid be given in cubical inches, it will be easy to reduce the same to the common liquid or dry measures, and conversely to reduce these to solid inches. The liquid and dry measures, in use among other nations, are known from their writers.

"As to the English liquid measures, by act of parliament 1706, any round vessel commonly called a cylinder, having an even bottom, being seven inches in diameter throughout, and six inches deep from the top of the inside to the bottom (which vessel will be found by computation to contain 230½ cubic inches), or any vessel containing 231 cubic inches, and no more, is deemed to be a lawful wine-gallon. An English pint therefore contains 28½ cubic inches; 2 pints make a quart; 4 quarts a gallon; 18 gallons a roundlet; 3 roundlets and an half, or 63 gallons, make a hogshead; the half of a hogshead is a barrel; 1 hogshead and a third, or 84 gallons, make a puncheon; 1 puncheon and a half, or 2 hogsheads, or 126 gallons, make a pipe or butt; the third part of a pipe, or 42 gallons, make a tierce; 2 pipes, or 3 puncheons, or 4 hogsheads, make a ton of wine. Though the English wine gallon is now fixed at 231 cubic inches, the standard kept in Guildhall being measured, before many persons of distinction, May 25, 1688, it was found to contain only 224 inch inches.

"In the English beer-measure, a gallon contains 282 cubic inches; consequently 35½ cubic inches make a pint, 2 pints make a quart, 4 quarts make a gallon, 9 gallons a firkin, 4 firkins a barrel. In ale, 8 gallons make a firkin, and 32 gallons make a barrel. By an act of the first of William and Mary, 34 gallons is the barrel, both for beer and ale, in all places, except within the weekly bill of mortality.

"In Scotland it is known that 4 gills make a mucklekin, 2 mucklekins make a chopin; a pint is two chopins; a quart is two pints; and a gallon is four quarts, or eight pints. The accounts of the cubical inches contained in the Scots pint vary considerably from each other. According to our author, it contains 109 cubical inches. But the standard jugs kept by the dean of guild of Edinburgh (one of which has the year 1555, with the arms of Scotland, and the town of Edinburgh, marked upon it) having been carefully measured several times, and by different persons, the Scots pint, according to those standards, was found to contain about 163½ cubic inches. The pewterers' jugs (by which the vessels in common use are made) are said to contain sometimes betwixt 105 and 106 cubic inches. A cask that was measured by the brewers of Edinburgh, before the commissioners of ex- Gauging.

cile in 1707, was found to contain 46\(\frac{3}{4}\) Scots pints; the same vessel contained 18\(\frac{3}{4}\) English ale-gallons. Supposing this mensuration to be just, the Scots pint will be to the English ale-gallon as 289 to 750; and if the English ale-gallons be supposed to contain 282 cubical inches, the Scots pint will contain 108.664 cubical inches. But it is suspected, on several grounds, that the experiment was not made with sufficient care and exactness.

"The commissioners appointed by authority of parliament to settle the measures and weights, in their act of Feb. 19, 1618, relate, That having caused fill the Linlithgow firlot with water, they found that it contained 21\(\frac{1}{2}\) pints of the just Stirling jug and measure. They likewise ordain that this shall be the just and only firlot; and add, That the widthens and breadths of the which firlot, under and above even over within the burids, shall contain nineteen inches and the sixth part of an inch, and the depthes seven inches and a third part of an inch. According to this act (supposing their experiment and computation to have been accurate) the pint contained only 99.56 cubical inches; for the content of such a vessel as is described in the act, is 2115.85, and this divided by 21\(\frac{1}{2}\) gives 99.56. But by the weight of water said to fill this firlot in the same act, the measure of the pint agrees nearly with the Edinburgh standard above mentioned.

"As for the English measures of corn, the Winchester gallon contains 27\(\frac{1}{2}\) cubical inches; 2 gallons make a peck; 4 pecks, or 8 gallons (that is, 2178 cubical inches), make a bushel; and a quarter is 8 bushels.

"Our author says, that 19\(\frac{1}{2}\) Scots pints make a firlot. But this does not appear to be agreeable to the statute above mentioned, nor to the standard-jugs. It may be conjectured, that the proportion assigned by him has been deduced from some experiment of how many pints, according to common use, were contained in the firlot. For if we suppose those pints to have been each of 108.664 cubical inches, according to the experiment made in the 1707 before the commissioners of excise, described above; then 19\(\frac{1}{2}\) such pints will amount to 2118.94, cubical inches; which agrees nearly with 2115.85, the measure of the firlot by statute above mentioned. But it is probable, that in this he followed the act 1587, where it is ordained, That the wheat-firlot shall contain 19 pints and two joucattes. A wheat-firlot marked with the Linlithgow stamps being measured, was found to contain about 2211 cubical inches. By the statute of 1618, the barley-firlot was to contain 31 pints of the just Stirling jug.

"A Paris pint is 48 cubical Paris inches, and is nearly equal to an English wine-quart. The Bolifian contains 664.68099 Paris cubical inches, or 780.36 English cubical inches.

"The Roman amphora was a cubical Roman foot, the congius was the eighth part of the amphora, the sextarius was one-sixth of the congius. They divided the sextarius like the as or libra. Of dry measures, the medimnus was equal to two amphoras, that is, about 14\(\frac{1}{2}\) English legal bushels; and the modius was the third part of the amphora."

PROPOSITION XLIII.

To find the solid content of a given prism.—By the 29th prop. let the area of the base of the prism be measured, and be multiplied by the height of the prism, the product will give the solid content of the prism.

PROPOSITION XLIV.

To find the solid content of a given pyramid.—The area of the base being found (by the 30th prop.), let it be multiplied by the third part of the height of the pyramid, or the third part of the base by the height, the product will give the solid content, by 17th 12. Eucl.

COROLLARY.

If the solid content of a frustum of a pyramid is required, first let the solid content of the entire pyramid be found; from which subtract the solid content of the part that is wanting, and the solid content of the broken pyramid will remain.

PROPOSITION XLV.

To find the content of a given cylinder.—The area of the base being found by prop. 33, if it be a circle, and by prop. 35, if it be an ellipse (for in both cases it is a cylinder), multiply it by the height of the cylinder, and the solid contents of the cylinder will be produced.

COROLLARY.

Fig. 23. And in this manner may be measured the solid content of vessels and casks not much different from a cylinder, as ABCD. If towards the middle EF it be somewhat groser, the area of the circle of the base being found (by 33d prop.) and added to the area of the middle circle EF, and the half of their sum (that is, an arithmetical mean between the area of the base and the area of the middle circle) taken for the base of the vessel, and multiplied into its height, the solid content of the given vessel will be produced.

Note, That the length of the vessel, as well as the diameters of the base, and of the circle EF, ought to be taken within the staves; for it is the solid content within the staves that is sought.

PROPOSITION XLVI.

To find the solid content of a given cone.—Let the area of the base (found by prop. 33) be multiplied into \(\frac{1}{3}\) of the height, the product will give the solid content of the cone; for by the 10th 12. Eucl. a cone is the third part of a cylinder that has the same base and height.

PROPOSITION XLVII.

Fig. 24. 25. To find the solid content of a frustum of a cone cut by a plane parallel to the plane of the base.—First, let the height of the entire cone be found, and thence (by the preceding prop.) its solid content; from which subtract the solid content of the cone cut off at the top, there will remain the solid content of the frustum of the cone.

How the content of the entire cone may be found, appears thus: Let ABCD be the frustum of the cone (either right or scaleneous, as in the figures 2. and 3.) let the cone ECD be supposed to be completed; let AG be drawn parallel to DE, and let AH and EF be perpendicular on CD; it will be (by 2d 6. Eucl.) as CG : CA :: CD : CE; but (by art. 72. of Part I.) as CA : AH :: CE : EF; consequently (by 22d 5. Eucl.) as \( CG : AH :: CD : EF \); that is, as the excess of the diameter of the lesser base is to the height of the frustum, so is the diameter of the greater base to the height of the entire cone.

**COROLLARY.**

Fig. 26. Some casks whose staves are remarkably bended about the middle, and strait towards the ends, may be taken for two portions of cones, without any considerable error. Thus \( ABEF \) is a frustum of a right cone, to whose base \( EF \), on the other side, there is another similar frustum of a cone joined, \( EDCF \). The vertices of these cones, if they be supposed to be completed, will be found at \( G \) and \( H \). Whence (by the preceding proposition) the solid content of such vessels may be found.

**PROPOSITION XLVIII.**

Fig. 27. A cylinder circumscribed about a sphere, that is, having its base equal to a great circle of the sphere, and its height equal to the diameter of the sphere, is to the sphere as 3 to 2.

Let \( ABEC \) be the quadrant of a circle, and \( ABDC \) the circumscribed square; and likewise the triangle \( ADC \); by the revolution of the figure about the right line \( AC \), as axis, a hemisphere will be generated by the quadrant, a cylinder of the same base and height by the square, and a cone by the triangle. Let these three be cut any how by the plane \( HF \), parallel to the base \( AB \); the section in the cylinder will be a circle whose radius is \( FH \), in the hemisphere a circle of the radius \( EF \), and in the cone a circle of the radius \( GF \).

By art. 69. of Part I. \( EAq \), or \( HFq = EFq \) and \( FAq \) taken together (but \( AFq = FGq \), because \( AC = CD \)); therefore the circle of the radius \( FH \) is equal to a circle of the radius \( EF \), together with a circle of the radius \( GF \); and since this is true everywhere, all the circles together described by the respective radii \( HF \) (that is, the cylinder) are equal to all the circles described by the respective radii \( EF \) and \( FG \) (that is, to the hemisphere and the cone taken together); but (by the 10th Eucl.) the cone generated by the triangle \( DAC \) is one third part of the cylinder generated by the square \( BC \). Whence it follows, that the hemispheroid generated by the rotation of the quadrant \( ABEC \) is equal to the remaining two third parts of the cylinder, and that the whole sphere is \( \frac{2}{3} \) of the double cylinder, circumscribed about it.

This is that celebrated 39th prop. 1. book of Archimedes of the sphere and cylinder; in which he determines the proportion of the cylinder to the sphere inscribed to be that of 3 to 2.

**COROLLARY.**

Hence it follows, that the sphere is equal to a cone whose height is equal to the semidiameter of the sphere, having for its base a circle equal to the superficies of the sphere, or to four great circles of the sphere, or to a circle whose radius is equal to the diameter of the sphere (by prop. 41. of this.) And indeed a sphere differs very little from the sum of an infinite number of cones that have their bases in the surface of the sphere, and their common vertex in the centre of the sphere; so that the superficies of the sphere (of whose dimension see prop. 41. of this) multiplied into the third part of the semidiameter, gives the solid content of the sphere.

---

**PROPOSITION XLIX.**

Fig. 28. To find the solid content of a sector of the sphere.—A spherical sector \( ABC \) (as appears by the corollary of the preceding prop.) is very little different from an infinite number of cones, having their bases in the superficies of the sphere \( BEC \), and their common vertex in the centre. Wherefore the spherical superficies \( BEC \) being found (by prop. 42. of this), and multiplied into the third part of \( AB \) the radius of the sphere, the product will give the solid content of the sector \( ABC \).

**COROLLARY.**

It is evident how to find the solidity of a spherical segment less than a hemispherical, by subtracting the cone \( ABC \) from the sector already found. But if the spherical segment be greater than a hemisphere, the cone corresponding must be added to the sector, to make the segment.

**PROPOSITION L.**

Fig. 29. To find the solidity of the spheroid, and of its segments cut by planes perpendicular to the axis.

In prop. 44. of this, it is shown, that every where \( EH : EG :: CF : CD \); but circles are as the squares described upon their rays, that is, the circle of the radius \( EH \) is to the circle of the radius \( EG \), as \( CFq \) to \( CDq \). And since it is so everywhere, all the circles described with the respective rays \( EH \) (that is, the spheroid made by the rotation of the semi-ellipses \( AFB \) around the axis \( AB \)) will be to all the circles described by the respective radii \( EG \) (that is, the sphere described by the rotation of the semicircle \( ADB \) on the axis \( AB \)) as \( FCq \) to \( CDq \); that is, as the spheroid to the sphere on the same axis, so is the square of the other axis of the generating ellipse to the square of the axis of the sphere.

And this holds, whether the spheroid be found by a revolution around the greater or lesser axis.

**COROLLARY 1.**

Hence it appears, that the half of the spheroid, formed by the rotation of the space \( AHFC \) around the axis \( AC \), is double of the cone generated by the triangle \( AFC \) about the same axis; which is the 32d prop. of Archimedes of conoids and spheroids.

**COROLLARY 2.**

Hence, likewise, is evident the measure of segments of the spheroid cut by planes perpendicular to the axis. For the segment of the spheroid made by the rotation of the space \( ANHE \), round the axis \( AE \), is to the segment of the sphere having the same axis \( AC \), and made by the rotation of the segment of the circle \( AMGE \), as \( CFq \) to \( CDq \).

But if the measure of this solid be wanted with less labour, by the 34th prop. of Archimedes of conoids and spheroids, it will be as \( BE \) to \( AC + EB \); so is the cone generated by the rotation of the triangle \( AHE \) round the axis \( AE \), to the segment of the sphere made by the rotation of the space \( ANHE \) round the same axis \( AE \); which could easily be demonstrated by the method of indivisibles.

**COROLLARY 3.**

Hence it is easy to find the solid content of the segment of a sphere or spheroid intercepted between two parallel planes, perpendicular to the axis. This agrees as well to the oblate as to the oblong spheroid; as is obvious. COROLLARY 4.

Fig. 30. If a cask is to be valued as the middle piece of an oblong spheroid, cut by the two planes DC and FG, at right angles to the axis: first, let the solid content of the half spheroid ABCED be measured by the preceding prop. from which let the solidity of the segment DEC be subtracted, and there will remain the segment ABCD; and this doubled will give the capacity of the cask required.

The following method is generally made use of for finding the solid content of such vessels. The double area of the greatest circle, that is, of that which is described by the diameter AB at the middle of the cask, is added to the area of the circle at the end, that is, of the circle DC or FG (for they are usually equal), and the third part of this sum is taken for a mean base of the cask; which therefore multiplied into the length of the cask OP, gives the content of the vessel required.

Sometimes vessels have other figures, different from those we have mentioned; the easy methods of measuring which may be learned from those who practise this art. What hath already been delivered is sufficient for our purpose.

PROPOSITION LI.

Fig. 31. and 32. To find how much is contained in a vessel that is in part empty, whose axis is parallel to the horizon.—Let AGBH be the great circle in the middle of the cask, whose segment GBH is filled with liquor, the segment GAH being empty; the segment GBH is known, if the depth EB be known, and EH a mean proportional between the segments of the diameter AB and EB; which are found by a rod or ruler put into the vessel at the orifice. Let the basis of the cask at a medium be found, which suppose to be the circle CKDL; and let the segment KCL be similar to the segment GAH (which is either found by the rule of three, because as the circle AGBH is to the circle CKDL, so is the segment GAH to the segment KCL; or is found from the tables of segments made by authors); and the product of this segment multiplied by the length of the cask will give the liquid content remaining in the cask.

PROPOSITION LII.

To find the solid content of a regular and ordinate body.

—A tetraedron being a pyramid, the solid content is found by the 44th prop. The hexaedron, or cube, being a kind of prism, it is measured by the 43d prop. An octaedron consists of two pyramids of the same square base, and of equal heights; consequently its measure is found by the 44th prop. A dodecaedron consists of 12 pyramids having equal equilateral and equiangular pentagonal bases; and so one of these being measured (by the 44th prop. of this), and multiplied by 12, the product will be equal to the solid content of the dodecaedron. The icosaedron consists of 20 equal pyramids having triangular bases; the solid content of one of which being found (by the 44th prop.), and multiplied by 20, gives the whole solid. The bases and heights of these pyramids, if you want to proceed more exactly, may be found by trigonometry. See Trigonometry.

PROPOSITION LIII.

To find the solid content of a body however irregular.

—Let the given body be immersed into a vessel of water, having the figure of a parallelopipedon or prism, and let it be noted how much the water is raised upon the immersion of the body. For it is plain, that the space which the water fills, after the immersion of the body, exceeds the space filled before its immersion, by a space equal to the solid content of the body, however irregular. But when this excess is of the figure of a parallelopipedon or prism, it is easily measured by the 43d prop. of this, viz. by multiplying the area of the base, or mouth of the vessel, into the difference of the elevations of the water before and after immersion: Whence is found the solid content of the body given.

In the same way the solid content of a part of a body may be found, by immersing that part only in water.

There is no necessity to insist here on diminishing or enlarging solid bodies in a given proportion. It will be easy to deduce these things from the 11th and 12th books of Euclid.

"The following rules are subjoined for the ready computation of contents of vessels, and of any solids in the measures in use in Great Britain.

"I. To find the content of a cylindric vessel in English wine gallons, the diameter of the base and altitude of the vessel being given in inches and decimals of an inch.

"Square the number of inches in the diameter of the vessel; multiply this square by the number of inches in the height: then multiply the product by the decimal fraction .0034; and this last product shall give the content in wine-gallons and decimals of such a gallon. To express the rule arithmetically; let D represent the number of inches and decimals of an inch in the diameter of the vessel, and H the decimals of an inch in the height of the vessel; then the content in wine-gallons shall be DDH×.0034, or DDH×.0034. Ex. Let the diameter D=51.2 inches, the height H=62.3 inches, then the content shall be 51.2×51.2×62.3×.0034 = 555.27.332 wine-gallons. This rule follows from prop. 33. and 45. For by the former, the area of the base of the vessel is in square inches DD×.7854; and by the latter, the content of the vessel in solid inches is DDH×.7854; which divided by 231 (the number of cubical inches in a wine-gallon) gives DDH×.0034, the content in wine gallons. But though the charges in the excise are made (by statute) on the supposition that the wine-gallon contains 231 cubical inches; yet it is said, that in sale 224 cubical inches, the content of the standard measured at Guildhall (as was mentioned above), are allowed to be a wine-gallon.

"II. Supposing the English ale-gallon to contain 282 cubical inches, the content of a cylindric vessel is computed in such gallons, by multiplying the square of the diameter of a vessel by its height as formerly, and their product by the decimal fraction .0027.851; that is, the solid content in ale-gallons is DDH×.0027.851.

"III. Supposing the Scots pint to contain about 103.4 cubical inches (which is the measure given by the standards at Edinburgh, according to experiments mentioned above), the content of a cylindric vessel is computed in Scots pints, by multiplying the square of the diameter of the vessel by its height, and the product... Gauging. duct of these by the decimal fraction .0076. Or the content of such a vessel in Scots pints is DDH × 0.0076.

"Supposing the Winchester bushel to contain 2187 cubical inches, the content of a cylindric vessel is computed in those bushels by multiplying the square of the diameter of the vessel by the height, and the product by the decimal fraction .0003666. But the standard bushel having been measured by Mr Everard and others in 1696, it was found to contain only 2145.6 solid inches; and therefore it was enacted in the act for laying a duty upon malt, That every round bushel, with a plain and even bottom, being 18½ inches diameter throughout, and 8 inches deep, should be deemed a legal Winchester bushel. According to this act (ratified in the first year of queen Anne) the legal Winchester bushel contains only 2150.42 solid inches. And the content of a cylindric vessel is computed in such bushels, by multiplying the square of the diameter by the height, and their product by the decimal fraction .0003666. Or the content of the vessel in those bushels is DDH × 0.003666.

"V. Supposing the Scots wheat-firlot to contain 21½ Scots pints (as is appointed by the statute 1618), and the pint to be conform to the Edinburgh standards above mentioned, the contents of a cylindric vessel in such firlots is computed by multiplying the square of the diameter by the height, and their product by the decimal fraction .00358. This firlot, in 1426, is appointed to contain 17 pints; in 1457, it was appointed to contain 18 pints; in 1587, it is 19½ pints; in 1628, it is 21½ pints: and though this last statute appears to have been founded on wrong computations in several respects, yet this part of that act that relates to the number of pints in the firlot seems to be the least exceptionable; and therefore we suppose the firlot to contain 21½ pints of the Edinburgh standard, or about 2197 cubical inches; which a little exceeds the Winchester bushel, from which it may have been originally copied.

"VI. Supposing the bear-firlot to contain 31 Scots pints (according to the statute 1618), and the pint conform to the Edinburgh standards, the content of a cylindric vessel in such firlots is found by multiplying the square of the diameter by the height, and this product by .00245.

"When the section of the vessel is not a circle, but an ellipsis, the product of the greatest diameter by the least is to be substituted in those rules for the square of the diameter.

"VII. To compute the content of a vessel that may be considered as a frustum of a cone in any of those measures.

"Let A represent the number of inches in the diameter of the greater base, B the number of inches in the diameter of the lesser base. Compute the square of A, the product of A multiplied by B, and the square of B, and collect these into a sum. Then find the third part of this sum, and substitute it in the preceding rules in the place of the square of the diameter; and proceed in all other respects as before. Thus, for example, the content in wine-gallons in \( \frac{AA \times AB \times BB}{3} \times H \times 0.0034 \).

"Or, to the square of half the sum of the diameters A and B, add one-third part of the square of half their difference, and substitute this sum in the preceding rules for the square of the diameter of the vessel; for the square of \( \frac{AA \times AB}{3} \times BB \), gives \( \frac{AA \times AB \times BB}{3} \).

"VIII. When a vessel is a frustum of a parabolic conoid, measure the diameter of the section at the middle of the height of the frustum; and the content will be precisely the same as of a cylinder of this diameter of the same height with the vessel.

"IX. When a vessel is a frustum of a sphere, if you measure the diameter of the section at the middle of the height of the frustum, then compute the content of a cylinder of this diameter of the same height with the vessel, and from this subtract \( \frac{1}{3} \) of the content of a cylinder of the same height on a base whose diameter is equal to its height; the remainder will give the content of the vessel. That is, if D represent the diameter of the middle section, and H the height of the frustum, you are to substitute DD - \( \frac{1}{3} \) HH for the square of the diameter of the cylindric vessel in the first six rules.

"X. When the vessel is a frustum of a spheroid, if the bases are equal, the content is readily found by the rule in p. 685. In other cases, let the axis of the solid be to the conjugate axis as n to 1; let D be the diameter of the middle section of the frustum, H the height or length of the frustum; and substitute in the first six rules DD - \( \frac{HH}{3n^2} \) for the square of the diameter of the vessel.

"XL. When the vessel is an hyperbolic conoid, let the axis of the solid be to the conjugate axis as n to 1, D the diameter of the section at the middle of the frustum, H the height or length; compute DD - \( \frac{HH}{3n^2} \), and substitute this sum for the square of the diameter of the cylindric vessel in the first six rules.

"XII. In general, it is usual to measure any round vessel, by distinguishing it into several frustums, and taking the diameter of the section at the middle of each frustum; thence to compute the content of each, as if it was a cylinder of that mean diameter; and to give their sum as the content of the vessel. From the total content, computed in this manner, they subtract successively the numbers which express the circular areas that correspond to those mean diameters, each as often as there are inches in the altitude of the frustum to which it belongs, beginning with the uppermost; and in this manner calculate a table for the vessel, by which it readily appears how much liquor is at any time contained in it, by taking either the dry or wet inches; having regard to the inclination or drip of the vessel when it has any.

"This method of computing the content of a frustum from the diameter of the section at the middle of its height, is exact in that case only when it is a portion of a parabolic conoid; but in such vessels as are in common use, the error is not considerable. When the vessel is a portion of a cone or hyperbolic conoid, the content by this method is found less than the truth; but when it is a portion of a sphere or spheroid, the content computed in this manner exceeds the truth. The difference or error is always the same in the different parts of the same or of similar vessels, when the altitude of the frustum is given. And when the altitudes are different, the error is in the triplicate ratio..." Gauging.

If exactness be required, the error in measuring the frustum of a conical vessel in this manner is \( \frac{1}{3} \) of the content of a cone similar to the vessel, of an altitude equal to the height of the frustum. In a sphere, it is \( \frac{1}{3} \) of a cylinder of a diameter and height equal to the frustum. In the spheroid and hyperbolic conoid, it is the same as in a cone generated by the right-angled triangle, contained by the two semiaxes of the figure, revolving about that side which is the semiaxis of the frustum.

In the usual method of computing a table for a vessel, by subducting from the whole content the number that expresses the uppermost area as often as there are inches in the uppermost frustum, and afterwards the numbers for the other areas successively; it is obvious, that the contents assigned by the table, when a few of the uppermost inches are dry, are stated a little too high if the vessel stands upon its base, but too low when it stands on its greater base; because, when one inch is dry, for example, it is not the area at the middle of the uppermost frustum, but rather the area at the middle of the uppermost inch, that ought to be subducted from the total content, in order to find the content in this case.

XIII. To measure round timber: Let the mean circumference be found in feet and decimals of a foot; square it; multiply this square by the decimal .079577, and the product by the length. Ex. Let the mean circumference of a tree be 10 feet, and the length 24 feet. Then \( 10^2 \times 0.079577 \times 24 = 202.615 \), is the number of cubical feet in the tree. The foundation of this rule is, that when the circumference of a circle is 1, the area is \( 0.795774 \times 715 \), and that the areas of circles are as the squares of their circumferences.

But the common way used by artificers for measuring round timber, differs much from this rule. They call one fourth part of the circumference the girt, which is by them reckoned the side of a square, whose area is equal to the area of the section of the tree; therefore they square the girt, and then multiply by the length of the tree. According to their method, the tree of the last example would be computed at 159.13 cubical feet only.

How square timber is measured, will be easily understood from the preceding propositions. Fifty solid feet of hewn timber, and forty of rough timber, make a load.

XIV. To find the burden of a ship, or the number of tons it will carry, the following rule is commonly given. Multiply the length of the keel taken within board, by the breadth of the ship within board, taken from the midship beam from plank to plank, and the product by the depth of the hold, taken from the plank below the keelson to the under part of the upper deck plank, and divide the product by 94, the quotient is the content of the tonnage required. This rule, however, cannot be accurate; nor can one rule be supposed to serve for the measuring exactly the burden of ships of all sorts. Of this the reader will find more in the Memoirs of the Royal Academy of Sciences at Paris for the year 1721.

Our author having said nothing of weights, it may be of use to add briefly, that the English Troy-pound contains 12 ounces, the ounce 20 penny-weight, and the penny-weight 24 grains; that the Averdupois pound contains 16 ounces, the ounce 16 drams, and that 112 pounds is usually called the hundred weight. It is commonly supposed, that 14 pounds Averdupois are equal to 17 pounds Troy. According to Mr Everard’s experiments, 1 pound Averdupois is equal to 14 ounces 12 penny-weight and 16 grains Troy, that is, to 7000 grains; and an Averdupois ounce is 437\(\frac{1}{2}\) grains. The Scots Troy-pound (which, by the statute 1718, was to be the same with the French,) is commonly supposed equal to 15\(\frac{1}{2}\) ounces English Troy, or 7560 grains. By a mean of standards kept by the dean of guild at Edinburgh, it is 7599\(\frac{1}{2}\) or 7600 grains. They who have measured the weights which were sent from London after the union of the kingdoms to be the standards by which the weights in Scotland should be made, have found the English Averdupois pound (from a medium of the several weights) to weigh 7000 grains, the same as Mr Everard; according to which, the Scots, Paris, or Amsterdam pound, will be to the pound Averdupois as 38 to 35. The Scots Troy-stone contains 16 pounds, the pound 2 marks or 16 ounces, an ounce 16 drops, a drop 36 grains. Twenty Scots ounces make a Tron-pound; but because it is usual to allow one to the score, the Tron-pound is commonly 21 ounces. Sir John Skene, however, makes the Tron-stone to contain only 19\(\frac{1}{2}\) pounds.”

GEORGE I. II. and III. kings of Great Britain.

George I., the son of Ernest Augustus, duke of Brunswick Lunenburgh, and elector of Hanover; succeeded to the throne of Great Britain in 1714, in virtue of an act of parliament, passed in the latter part of the reign of king William III., limiting the succession of the crown, after the demise of that monarch, and queen Anne (without issue), to the princess Sophia of Hanover, and the heirs of her body, being Protestants.—George II., the only son of the former, succeeded him in 1727, and enjoyed a long reign of glory; dying amidst the most rapid and extensive conquests in the 77th year of his age. He was succeeded by his grandson George III., our present sovereign. For particulars, see BRITAIN, no. 374—701.