Hypograph (mathematics)

a page from "On the Sphere and Cylinder" in latin

On the Sphere and Cylinder is a work that was published by Archimedes in two volumes c. 225 BC. It most notably details how to find the surface area of a sphere and the volume of the contained ball and the analogous values for a cylinder, and was the first to do so.

Contents

Volume of Sphere to Volume of Cylinder is 2 to 3

The principal formulae derived in On the Sphere and Cylinder are those mentioned above: the surface area of the sphere, the volume of the contained ball, and surface area and volume of the cylinder. In his work, Archimedes showed that the surface area of a cylinder is equal to:

$A_{C}=2\pi r^{2}+2\pi rh=2\pi r(r+h).\,$

and that the volume of the same is:

$V_{C}=\pi r^{2}h.\,$

On the sphere, he showed that the surface area is four times the area of its great circle. In modern terms, this means that the surface area is equal to:

$A_{S}=4\pi r^{2}.\,$

The result for the volume of the contained ball stated that it is two-thirds the volume of a circumscribed cylinder, meaning that the volume is

$V_{S}={\frac {4}{3}}\pi r^{3}.$

Archimedes was particularly proud of this latter result, and so he asked for a sketch of a sphere inscribed in a cylinder to be inscribed on his grave - since h=2r, he had shown that both the volume and the surface area of the sphere were two-thirds that of the cylinder. Later, Roman philosopher Marcus Tullius Cicero discovered the tomb, which had been overgrown by surrounding vegetation.

The argument Archimedes used to prove the formula for the volume of a ball was rather involved in its geometry, and many modern textbooks have a simplified version using the concept of a limit, which, of course, did not exist in Archimedes' time. Archimedes used an inscribed half-polygon in a semicircle, then rotated both to create a conglomerate of frustums in a sphere, of which he then determined the volume.

It seems that this is not the original method Archimedes used to derive this result, but the best formal argument available to him in the Greek mathematical tradition. His original method likely involved a clever use of levers. A palimpsest rediscovered in the 19th century contained many of Archimedes works, including The Method of Mechanical Theorems, in which he describes a method to determine volumes which involves balances, centers of mass and infinitesimal slices.

Notes

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References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• S.H. Gould, The Method of Archimedes, The American Mathematical Monthly. Vol. 62, No. 7 (Aug. - Sep., 1955), pp. 473-476

30 year-old Entertainer or Range Artist Wesley from Drumheller, really loves vehicle, property developers properties for sale in singapore singapore and horse racing. Finds inspiration by traveling to Works of Antoni Gaudí.

• Lucio Lombardo Radice, La matematica da Pitagora a Newton, Roma, Editori Riuniti, 1971.
• Attilio Frajese, Opere di Archimede, Torino, U.T.E.T., 1974.