BİRİNCİ FEN · BİRİNCİ BÂB · DÖRDÜNCÜ FASIL · Dördüncü Madde
Fourth Article
It pertains to the areas of surfaces (formed figures), the volumes of solids, and the heights of objects whose height cannot be measured under normal circumstances.
O esteemed one! Know that geometers have stated: The quantity of a surface is its area. Thus, calculating the measurement of a surface occurs by knowing its surface measure.
**Calculations of Area**
The area of a right-angled triangle (ką’imü’z-zâviye) is obtained by multiplying one of the sides forming the right angle by half of the other side. The area of an obtuse triangle (münfericü’z-zâviye) is either half the product of the perpendicular drawn from the vertex of the obtuse angle to the opposite side and that opposite side, or alternatively, the product of half the perpendicular and the opposite side. The area of an acute triangle (hâddü’z-zâviye) is equal to either the product of a perpendicular drawn from any vertex to the opposite side and half of that side, or alternatively, the product of half the perpendicular and the opposite side. The area of an equilateral triangle (mütesâviyü’l-ıdlâ’) is found by taking the square root of the product of one side with one-fourth of the square of that side.
The area of a square (murabba‘) is obtained by multiplying one side by itself. The area of a rectangle (mustatîl) is obtained by multiplying the length of one side by the length of an adjacent side. The area of a rhombus (şekl-i muayyen) is found by multiplying half of one diagonal by the entirety of the other diagonal. Similarly, the area of a parallelogram (şibh-i muayyen) is also obtained by multiplying half of one diagonal by the entirety of the other diagonal. The areas of polygons with an even number of sides, such as a regular hexagon and a regular octagon, are found by multiplying half the perimeter by half the diagonal. The diameters of all polygonal figures are found by joining the midpoints of two opposite sides with a straight line.
The area of a circle is obtained by drawing a string around it and multiplying half the length of the string by the radius. If the circumference of a circle is multiplied by 3 and one-seventh, the diameter is obtained; in this case, there is no need for a string. If the circumference of a circle is divided by 3 and one-seventh, the diameter emerges. For every circle’s circumference is the multiple of its diameter by three and one-seventh (Pi number). Therefore, if a circle's diameter is multiplied by 22 and the result is divided by 7, the resulting value is the circumference of that circle. And if the circumference of a circle is multiplied by 7 and the product is divided by 22, the quotient is the diameter of that circle. The area of a cube (müka‘‘ab) is known from the measurement of its side.
The area of a right cylinder (üstüvâne) is equal to the product of the circumference of its base and its height. The area of a cone (mahrût-ı kãim) is obtained by multiplying half the circumference of the base by the perpendicular drawn from the apex to the center of the base. The base areas of cones and cylinders are calculated in the same manner as the area of a circle. The surface area of a sphere is obtained by multiplying its diameter by the area of its great circle.
The total surface area of a sphere is found by multiplying the radius cubed by the area of a triangle. Or, it is possible to obtain the entire quantity by subtracting seven and half of seven from the cube of the sphere’s diameter, then removing the deficiency from the remaining amount in the same manner. By means of these, the volumes and distances of the celestial spheres and stars can be calculated.
A simple way to calculate the heights of tall objects is as follows: If it is possible to reach the lowest point of a tall object located on a flat surface; erect a pole and retreat until you can see the top point of the object from over the pole. Then, measure with steps or another instrument from your location to the base of the tall object, and multiply that total length by the height of the pole. Next, divide the distance you are from the object by the distance between your location and the object, and add your height to the result. The result is the height of the object.
Another way to calculate the heights of tall objects is as follows: Place a mirror on level ground near the base of the tall object. Retreat until you can see the top point of the tall object in the mirror. Multiply the distance between the mirror and the tall object by your own height. Divide the resulting product by the distance between your location and the mirror.
The result is the height of the aforementioned tall object. For the ratio of your height to the distance between you and the mirror is equal to the ratio of the distance between the mirror and the top of the tall object. Thus, it is one of the extremes in a proportional four-number sequence. For in a proportional four-number sequence, your height is the first number, the distance between you and the mirror is the second number, the height of the tall object is the third number, and the distance between the mirror and the base of the tall object is the fourth number. In this case, the height of the tall object is the unknown number. If you multiply the inner numbers and divide by the outer numbers, the result will be the unknown number.
Another way is as follows: Erect a stick, find the ratio of its shadow to itself. Then, find the height of the tall object from its shadow. When the sun rises 45 degrees above the horizon, every object’s shadow is equal to its own height. In geometry, we suffice with this much. After that, in accordance with the verse “Do they not reflect on the dominion of the heavens and the earth, so that they may be warned?” (Al-A'raf 7:185), we can begin writing a little about the position of the universe (ilm-i hey’et) to help one recognize the Sublime Essence, the ultimate goal.
The area of a cube is six times the area of one of its constituent squares.
The area of a circle is obtained by drawing a string around it and multiplying half the length of the string by the radius.
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The circumference of a circle is obtained by multiplying its diameter by three and one-seventh; in this case, rope is unnecessary. If the circumference of a circle were divided by three and one-seventh, the result would be its diameter. For every circle’s circumference is the multiple of its diameter by three and one-seventh (Pi). Therefore, if the diameter were multiplied by twenty-two, and the product were divided by seven, the result would be the circle's circumference. And if the circumference were multiplied by seven, and the product were divided by twenty-two, the quotient would be that circle’s diameter.
The area of a cube is known from the measurement of its side.
The area of a cylinder:
[29/b]
[30/a]
[30/b]
It explains, with a reasoned method across ten sections, the proof of the spherical shape of the universe and the details of the condition of stars and celestial spheres.