Also indicate the scales taken along both the axes on the graph paper. Time should not be wasted in taking a very large number of observations. The observations must be covering all available range evenly. Then try to guess where there will be sharp changes in the curvature of the curve. Take more readings in those regions. The size of the spread of plotted point must be in accordance with the accuracy of the data.
The central dot is the value of measured data. If the circle radius is large, it will mean as if uncertainty in data is more. Further such a representation tells that accuracy along x- and y-axis are the same. In case, uncertainty along the x-axis and y-axis are different, some of the notations used are accuracy along x-axis is more than that on y-axis ; accuracy along x-axis is less than that on y-axis.
You can design many more on your own. Now a days computers are also used for plotting graphs of a given data. This is indicated in Fig. Further, the slope of a given straight line has the same value, for all points on the line.
It is because the value of y changes by the same amount for a given change in the value of x, at every point of the straight line, as shown in Fig. Thus, for a given straight line, the slope is fixed. While calculating the slope, always choose the x-segment of sufficient length and see that it represents a round number of the variable. The corresponding interval of the variable on y-segment is then measured and the slope is calculated.
Generally, the slope should not have more than two significant digits. The values of the slope and the intercepts, if there are any, should be written on the graph paper.
Also keep in mind that slope of a graph has physical significance, not geometrical. Often straight-line graphs expected to pass through the origin are found to give some intercepts.
Hence, whenever a linear relationship is expected, the slope should be used in the formula instead of the mean of the ratios of the two quantities. However, it is not true for a curve. As shown in Fig. The slope of the curve at a particular point, say point A in Fig.
As such, in order to find the slope of a curve at a given point, one must draw a tangent to the curve at the desired point. In order to draw the tangent to a given curve at a given point, one may use a plane mirror strip attached to a wooden block, so that it stands perpendicular to the paper on which the curve is to be drawn.
This is illustrated in Fig. The slope of the tangent GAH i. The above procedure may be followed for finding the slope of any curve at any given point.
The students should thoroughly understand the principle of the experiment. The objective of the experiment and procedure to be followed should be clear before actually performing the experiment. The apparatus should be arranged in proper order.
To avoid any damage, all apparatus should be handled carefully and cautiously. Any accidental damage or breakage of the apparatus should be immediately brought to the notice of the concerned teacher. Precautions meant for each experiment should be observed strictly while performing it. Repeat every observation, a number of times, even if measured value is found to be the same.
The student must bear in mind the proper plan for recording the observations. Recording in tabular form is essential in most of the experiments. Calculations should be neatly shown using log tables wherever desired.
The degree of accuracy of the measurement of each quantity should always be kept in mind, so that final result does not reflect any fictitious accuracy. The result obtained should be suitably rounded off. Wherever possible, the observations should be represented with the help of a graph.
Always mention the result in proper SI unit, if any, along with experimental error. Also, write the formula used, explaining clearly the symbols involved derivation not required. Mention clearly, on the top of the observation table, the least counts and the range of each measuring instrument used. However, if the result of the experiment depends upon certain conditions like temperature, pressure etc.
Calculate experimental error. Wherever possible, use the graphical method for obtaining the result. To find the solidity of a frustum of a pyramid. Add together the areas of the two bases, and a mean proportional between them, and multiply the sum by one third of the altitude. When the pyramid is regular, it is generally most convenient to find the area of its base by Rule II. If we put a to represent one side of the lower base, and b one side of the upper base, andcthe tabular number from Art.
What is the solidity of a frustum of an hexagonal pyramid whose altitude is 15 feet, each side of the greater end being 3 feet, and that of the less end 2 feet? What is the solidity of a frustum of an octagonal pyramid whose altitude is 9 feet, each side of the greater end being 30 inches, and that of the less end 20 inches? A wedge is a solid bounded by five planes, viz. To find the solidity of a wedge.
Add the length of the edge to twice the length of the base, and multiply the sum by one sixth of the product of the height of the wedge and the breadth of the base.
Now, if the length of the base is equal to that of the edge,. The wedge will be divided into two parts, viz. The solidity of the former is equal to bhl L-1 , and that of the latter is Ibhl. What is the solidity of a wedge whose base is 30 inches long and 5 inches broad, its altitude 12 inches, and the length of the edge 2 feet? What is the solidity of a wedge whose base is 40 inches long and 7 inches broad, its altitude 18 inches, and the length of the edge 30 inches?
A rectangular prismoid is a solid bounded by six planes, of which the two bases are rectangles having their corresponding sides parallel, and the four upright sides of the sol.
To find the solidity of a rectangular prismoid. Add together the areas of the two bases, and four times the area of a parallel section equally distant from the bases, and multiply the sum by one sixth of the altitude. What are the contents of a log of wood, in the form of a rectangular prismoid, the length and breadth of one end being 16 inches and 12 inches, and of the other 7 inches and 4 inches, the length of the log being 24 feet?
What is the solidity of a log of hewn timber, whose ends are 18 inches by 15, and 14 inches by , its length being 18 feet? To compute the excavation or embankment for a rail-way. By the preceding rule may be computed the amount of excavation or embankment required in constructing a railroad or canal. If we divide the line of the road into portions so small that each may be regarded as a straight line, and suppose an equal number of transverse sections to be made, the excavation or embankment between two sections may be regarded as a prismoid, and its contents found by the pre ceding rule.
Let ABCD represent the lower surface of the supposed excavation, which we will assume to be parallel to the horizon: and let EFGH represent the upper surface of the excavation.
The A' 1' parts upon each side of the middle prismoid are also halves of rectangular prismoids; or, if the two parts are equal, they may be regarded as constituting a second prismoid, one of whose bases is the sum of the triangles A'E'I, B'F'K; and the other base is the sum of the triangles C'G'L, D'H'IM.
Therefore the volume of the entire solid is equal to the product of one sixth of its length, by the sum of the areas of the sections at the two extremities, and four times the area of a parallel and equidistant section. The annexed figure repre- c d e sents a cross section, showing the form of the excavation. The base of the cutting is to a b.
Calculation of the portion ABH. Since BH is 18 feet, the length of cd in the cross section will be 27 feet, and cf, the breadth at the top of the section, will be feet.
We accordingly find, by Art. For the middle section,the height is 3 feet, cd is Calculation of the portion BCIH. Since CI is 20 feet, the length of cd is 30 feet, and cf is I tO feet. Calculation of the portion CID. The height of the middle section is 10 feet; therefore cf is 80 feet, and the area of the cross section is.
Calculation of the portion DKE.. Since EK is 19 feet, the length of cd is 38 feet, and cf is feet. Calculation of the portion KEFL. Since LF is 8 feet, cd is 16 feet, and cf is 82 feet. The area of the section at LF is therefore equal to. Calculation of the portion LFG. The base of the cutting to be 50 feet wide, and the slope j1 horizontal to 1 perpendicular.
To find the surface of a regular polyedron. Multiply the area of one of the faces by the number of F. Since all the faces of a regular polyedron are equal, it is evident that the area of one of them, multiplied by their number, will give the entire surface.
Also, regular solids of the same name are similar, and similar polygons are as the squares of their homologous sides Geom. The following table shows the surface and solidity of regular polyedrons whose edge is unity. The surface is obtained by multiplying the area of one of the faces, as given in Art. Thus the area of an equilateral triangle, whose side is 1, is 0.
A Table of the regular Polyedrons whose Edges are unity Names. Tetraedron, 4 1. Hexaedron, 6 6. Octaedron, 8 3. Dodecaedron, 12 Icosaedron, 20 8. What is the surface of a regular octaedron whose edges are each 8 feet?
What is the surface of a regular dodecaedron whose edge is 12 feet? To find the solidity of a regular polyedron. Multiply the surface by one third of the perpendicular lei fall from the center on one of the faces; or, Multiply the cube of one of the edges by the solidity of a similar polyedron, whose edge is unity.
Since the faces of a regular polyedron are similar and equal,. If planes be made to pass through the center and the several edges of the solid, they will divide it into as many equal pyramids as it has faces.
The base of each pyramid will be one of the faces of the polyedron; and since their altitude is the perpendicular from the center upon one of the faces, the solidity of the polyedron must be equal to the areas of all the faces, multiplied by one third of this perpendicular. Also, similar pyramids are to each other as the cubes ot their homologous edges Geom. And since two regular polyedrons of the same name may be divided into the same number of similar pyramids, they must be to each other as the cubes of their edges.
The solidity of a tetraedron whose edge is unity, may be computed in the following manner: Let C-ABD be a tetraedron. Hence, in the right-angled triangle CEF, knowing one side and the angles, we can compute CE, which is found to be 0.
Whence, knowing the base ABD Art. In a somewhat similar manner may the solidities of the other regular polyedrons, given in Art.
What is the solidity of a regular tetraedron whose edges are each 24 inches? What is the solidity of a regular icosaedron whoso -dges are each 20 feet? To find the surface of a cylinder. Multiply the circumference of the base by the altitude fot the convex surface. What is the convex surface of a cylinder whose altitude is 23 feet, and the diameter of its base 3 feet? What is the entire surface of a cylinder whose alti.
To find the solidity of a cylinder. What is the solidity of a cylinder whose altitude is 18 feet 4 inches, and the diameter of its base 2 feet 10 inches? What is the solidity of a cylinder whose altitude is 12 feet 11 inches, and the circumference of its base 5 feet 3 inches? To find the surface of a cone. Multiply the circumference of the base by half the side for the convex surface; to which add the area of the base when the entire surface is required.
What is the entire surface of a cone whose side is 10 feet, and the diameter of its base 2 feet 3 inches? What is the entire surface of a cone whose side is 15 feet, and the circumference of its base 8 feet?
To find the solidity of a cone. IMultiply the area of the base by one third of the altitude. What is the solidity of a cone whose altitude is 12 feet, and the diameter of its base 21 feet?
What is the solidity of a cone whose altitude is 25 feet, and the circumference of its base 6 feet 9 inches? To find the surface of a frustum of a cone. Multiply half the side by the sum of the circumferences of the two bases for the convex surface; to this add the areas of the two bases when the entire surface is required.
What is the entire surface of a frustum of a cone, the diameters of whose bases are 9 feet and 5 feet, and whose side is 16 feet 9 inches? What is the convex surface of a frustum of a cone whose side is 10 feet, and the circumferences of its bases 6 feet and 4 feet? To find the solidity of a frustum of a cone. RULE Add together the areas of the two bases, and a mean pr.
If we put R and r for the radii of the two bases, then 7qrB will represent the area of one base, 7rr' the area of the other, and 7rRr the mean proportional between them. What is the solidity of a frustum of a cone whose altitude is 20 feet, the diameter of the greater end 5 feet, and that of the less end 2 feet 6 inches?
The length of a mast is 60 feet, its diameter at the greater end is 20 inches, and at the less end 12 inches: what is its solidity? To find the surface of a sphere. Multiply the diameter by the circumference of a great czrcle; or, Multiply the square of the diameter by 3. Required the surface of the earth, its diameter being miles. Required the surface of the moon, its circumference being miles. To find the solidity of a sphere. Multiply the surface by one third of the radius; or, Multiply the cube of the diameter by Ir; that is, by 0.
Where great accuracy is required, the value of -7r must be. Its value. What is the solidity of the earth, if it be a sphere miles in diameter? If the diameter of the moon be miles, what is its solidity? To find the surface of a spherical zone. Multiply the altitude of the zone by the circumference of a great circle of the sphere. If the diameter of the earth be miles, what is the surface of the torrid zone, extending 27' 36" on each side of the equator?
CAD: CD. On the same suppositions, find the surface of each of the temperate zones. To find the solidity of a spherical segment with one base. Multiply half the height of the segment by the area of the base, and the cube of the height by. The solidity of a spherical segment of two bases is the difference between two spherical segments, each having a single base. On the same supposition as in Ex. Find the solidity of the torrid zone.
To find the area of a spherical triangle. Compute the surface of the quadrantal triangle, or one eighth of the surface of the sphere. From the sum of the three angles subtract two right angles; divide the remainder by 90, and multiply the quotient by the quadrantal triangle See Geometry, Prop.
If the excess of the angles above two right angles is expressed in seconds, we must divide it by 90 degrees also expressed in seconds; that is, by , To find the area of a spherical polygon.
Compute the surface of the quadrantal triangle. From the sum of all the angles subtract the product of two right angles by the number of sides less two; divide the remainder by 90, and multiply the quotient by the quadrantal triangle. What is the area of a spherical polygon of 5 sides on a sphere whose diameter is 10 feet, supposing the sum of the angles to be degrees? Required the area of the polygon.
THE term Surveying includes the measurement ot heights and distances, the determination of the area of portions of the earth's surface, and their delineation upon paper. Since the earth is spherical, its surface is not a plane surface, and if large portions of the earth are to be rMeasured, the curvature must be taken into account; but in ordinary surveying, the portions of the earth are supposed to be so small that the curvature may be neglected.
The parts surveyed are therefore regarded as plane figures. If a plummet be freely suspended by a line, and allowed to come to a state of. Every plane passing through a vertical line is a vertical plane. A line perpendicular to a vertical line is a horizontal line. A plane perpendicular to a vertical line is a horizontal plane. A vertical angle is one the plane of whose sides is vertical. A horizontal angle is one the plane of whose sides is horizontal.
An angle of elevation is a vertical angle having one side horizontal and the other an ascending D line, as the angle BAD. An angle of depression is a vertical angle having one side horizontal and the other a descending line, as the angle CDA.
When distances are to be found A B by trigonometrical computation, it is necessary to measure at least one line upon the ground, and also as many angles as may be necessary to render three parts of every triangle known. This chain is divided into links. Sometimes a half chain is used, containing 50 links. To measure a horizontal line.
To mark the termination of the chain in measuring, ten iron pins should be provided, about a foot in length. Let the person who is to go foremost in carrying the chain, and who is called the leader, take one end of the chain and the ten pins; and let another person take the other end of the chain, and hold it at the beginning of the line to be measured.
When the leader has advanced until the chain is stretched tight, he must set down one pin at the end of the chain, the other person taking care that the chain is in the direction of the line to be measured. Then measure a second chain in the same manner, and so on until all the marking pins are exhausted. A record should then be made that ten chains have been measured, after which the marking pins should be returned to the leader, and the measurement continued as before until the whole line has been passed over.
It is generally agreed to refer all surfaces to a horizontal plane. Hence, when an inclined surface, like the side of a hill, is to be measured, the chain should be maintained in a horizontal position.
For this purpose, in ascending a hill, the hind end of the chain should be raised from the ground until it is on a level with the fore end, and should be held vertically over the termination of the preceding chain. In descending a hill, the fore end of the chain should be raised in the same manner. In measuring angles, some instrument is used which contains a portion of a graduated circle divided into degrees and minutes.
These instruments may be adapted to measuring. The instrument most fre. The principal parts of this instrument are a compassbox, a magnetic needle, two sights, and a stand for its support.
The compass-box, ABC, is circular, generally about six inches in diameter, and at its center is a small pin on which the magnetic needle is balanced. The circumference of the box is divided into degrees, and sometimes to half degrees; and the degrees are numbered from the extremities of a diameter both ways to The sights, DE, FG, are placed at right angles o the Xaofhgaun ir to the plane of the graduated circle, and in each of these there is a large and small aperture for convenience of observation The instrument, when used, is mounted on a tripod, or a single staff pointed with iron at the bottom, so that it may be firmly placed in the ground.
Sometimes two spirit levels, H and K, are attached, to indicate when the plane of the graduated circle is brought into a horizontal position. When the magnetic needle is supported so as to turn freely, and is allowed to come to a state of rest, the direction it assumes is called the magnetic meridian, one end of the needle indicating the north point and the other the south.
A horizontal line perpendicular to a meridian is an east and west line. The bearing or course of a line is the angle which it makes with a meridian passing through one end; and it is reckoned from the north or south point of the horizon, toward the east or west. B C The reverse bearing of a line is the bearing taken from the other end of the line.
The forward bearing and reverse bearing A of a line are equal angles, but lie between directly opposite points. It consists of a quarter of a circle, usually made of brass, and its limb, AB, is divided into A degrees and minutes, numbered c from A up to It is furnished either with a pair of plain sights or with a telescope, CD, which is to be directed toward I the object observed.
A plumb line, CE, is suspended from the center of the quadrant, and indicates when the radius CB is B brought into a vertical position. Move the telescope until the given object is seen in the middle of the field of view.
The center of the field is indicated by two wires placed in the focus of the object-glass of. When the horizontal wire is made to coincide with the summit of the tower, the angle of elevation is shown upon the are AB by means of an index which moves, with the telescope. As the arc is not commonly divided into parts smaller than half degrees, when great accuracy is required, some contrivance is needed for obtaining smaller fractions of a degree.
This is usually effected by a vernier. A Vernier is a scale of small extent, graduated in such a manner that, being moved by the side of a fixed scale, we are enabled to measure minute portions of this scale. The length of this movable scale is equal to a certain number of parts of that to be subdivided, but it is divided into parts one more or one less than those of the primary scale taken for the length of the vernier. Thus, if we wish to measure hundredths of an inch, as in the case of a barometer, we first divide an inch into ten equal parts.
We then construct a vernier equal in length to 11 of these divisions, but divide it into 10 equal parts, by which means each division on the vernier is T-'th longer than a division of the primary scale. Therefore, each division of Ulii i 3 the vernier is?! In practice, therefore, we ob. A similar contrivance is applied to graduated circles, to obtain the value of an arc with greater accuracy. If a circle is graduated to half degrees, or 30', and we wish to measure single minutes by the vernier, we take an arc equal to 31 divisions upon the limb, and divide it into 30 equal parts.
Then each division of the vernier will be equal to m ths of a degree, while each division of the scale is 3ths of a degree. That is, each space on the vernier exceeds one on the limb by 1'. In order, therefore, to read an angle for any position of the vernier, we pass along the vernier until a line is found coinciding with a line of the limb.
The number of this line from the zero point indicates the minutes which are to be added to the degrees and half degrees taken from the graduated circle. Sometimes a vernier is attached to the common surveyor's compass. The theodolite has two circular brass plates, C and D see fig.
Both have a horizontal motion about the vertical axis, E. This axis consists of two parts, one external, and the other internal; the former secured to the graduated limb, D, and the latter to the vernier plate, C, so that the vernier plate turns freely upon the lower.
The edge of the lower plate is divided into degrees and half degrees, and this is subdivided by a vernier attached to the upper plate into single minutes. The degrees are numbered from 0 to The parallel plates, A and B, are held together by a ball which rests in a socket. Four screws, three of which, a, a, a, are shown in the figure, turn in sockets fixed to the lower plate, while their heads press against the under side of the upper plate, by which means the instrument is leveled for observation.
The whole rests upon a tripod, which is firmly attached to the body of the instrument. To the vernier plate, two spirit-levels, c, c, are attached at right angles to each other, to determine when the graduated limb is horizontal.
A compass, also, is placed at F. Two frames, one of which is seen at N, support the pivots of the horizontal axis of the vertical semicircle KL, on which the telescope, GH, is placed.
One side of the vertical arc is divided into degrees and half degrees, and it is divided into single minutes by the aid of its vernier. Under and parallel to the telescope is a spirit-level, M, to show when the telescope is brought to a horizontal position. To enable us to direct the telescope upon an object with precision, two lines called wires are fixed at right angles to each other in the focus of the telescope. To measure a Horizontal Angle with the Theodolite.
Place the instrument exactly over the station from which the angle is to be measured; then level the instrument by means of the screws, a, a, bringing the telescope over each pair alternately until the two spirit-levels on the vernier plate retain their position, while the instrument is turned entirely round upon its axis.
Direct the telescope to one of the objects. Now read off the degrees upon the graduated limb, and the minutes indicated by the vernier. Next, release the upper plate leaving the graduated limb undisturbed , and move it round until the telescope is directed to the second object, and make the cross-wires bisect this object, as was done by the first. Again, read off the vernier; the difference between this and the former reading will be the angle required.
The magnetic bearing of an object is determined by simply reading the angle pointed out by the compass-needle when the object is bisected. To measure an Angle of Elevation with the Theodolite.
Direct the telescope toward the given object so that it may be bisected by the horizontal wire, and then read off the arc upon the vertical semicircle. After observing the object with the telescope in its natural position, it is well to revolve the telescope in its supports until the level comes uppermost, and repeat the observation.
The mean of the two measures may be taken as the angle of elevation. By the aid of the instruments now described, we may determine the distance of an inaccessible object, and its height above the surface of the earth. To determine the height of a vertical object situatea on a horizontal plane.
Measure from the object to any convenient distance in a straight line, and then take the angle of elevation subtended by the object. What is the height of the tower? Required the height of the wall, and the length of a ladder necessary to reach from my station to the top of it. The height is Length of ladder, To find the distance of a vertical object whose height is known.
Measure the angle of elevation, and we shall have given the angles and perpendicular of a right-angled triangle to find the base Art. What was its dis- D tance? What was its distance? To find the height of a vertical object standing on an inclined plane. Measure the distance from the object to any convenient station, and observe the angles which the base-line makes with lines drawn from its two ends to the top of the object. Required the height of the object.
ACB: AB:: sin. A tower standing on the top of a declivity, I measured 75 feet from its base, and then took the angle BAC, 50'; going on in the same direction 40 feet further, I took the angle BDC, 30'. What was the height of the tower? Let C be the object inaccessible from A and B. Then, if the dis. What was the dis nce between each station and the house? Then sin.
What are the respective distances from the fort? To find the distance between two objects separated by an impassable barrier. Measure the distance from any convenient station to each of the objects, and the angle included between those lines. We shall then have given two sides of a triangle and the included angle to find the third side.
What is the distance of the places C and B? In order to find the distance between two objects, C -d B, which could not be directly measured, I measured I. What is the distance between the objects C and B? To find the height of an inaccessible object above a horizontal plane. First Method. What is the height of the spire, supposing the instrument to have been five feet above the ground at each observation? Let DC be the given object. Also, if we observe the angle of elevation CBI ,.
Required the height of the spire. Required the height of the castle. To find the distance between two inaccessible objects. Measure any convenient base-line, and the angles between this base and lines drawn from each of its extremities to each of the objects. Let C and D be the two inaccessible objects. What is the distance of the house from the mill? Wanting to know the distance between two inaccessible objects, C and D, I measured a base-line, AB, What is the distance from C to D?
The area or content of a tract of land is the horizontal surface included within its boundaries. When the surface of the ground is broken and uneven, it is very difficult to ascertain exactly its actual surface. Hence it has been agreed to refer every surface to a horizontal plane; and for this reason, in measuring the boundary lines, it is necessary to reduce them all to horizontal lines. Mathematicians make mathematics difficult.
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Post a Comment. How much is 2 to the power zero five? A trick question, like everything mathematicians come up with for us. In short, transformations look Pages Home New Math. The trigonometric table of values of sines and cosines within one minute is counted on blondes. For comfort of the use for sines and cosines distinguished this table of value of corners by different colors. For sines the blue color of cells is accepted with degrees and by minutes.
For cosines the green background of cells is accepted. A yellow background is distinguish the values of minutes that if necessary is added or subtracted from tabular values. Usually it is not accepted so in detail to give a navigation on a trigonometric table. Firstly, a table is counted on experience users by mathematics. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public.
We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. An easy method of constructing mathematical tables including the logarithms of numbers sines tangents secants versed sines and their application etc An easy method of constructing mathematical tables including the logarithms of numbers sines tangents secants versed sines and their application etc Philip GARRETT Mathematician Philip GARRETT Mathematician.
Patent Office. Library United States. Author : United States. Tables of correct and concise Logarithms for numbers sines tangents secants complements arithmetical supplements etc with a compendious introduction to Logarithmetic Tables of correct and concise Logarithms for numbers sines tangents secants complements arithmetical supplements etc with a compendious introduction to Logarithmetic Samuel DUNN Professor of Mathematics.
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