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Q: What was the surveying instrument George Washington used to precisely measure angles and and land boundaries?

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taking readings using theodolite keeping the vertical circle of the instrument on left side of the observer is called a face left observation and to measure horizontal angle the instrument is rotated(swing) in horizontal plane with respect to the vertical axis in a clockwise direction called rigt swing.

it is line set out by the optical axis of the instrument ( level). so it is just an imaginary line that describes the ray of light that allowes us to read different values from the leveling staff. this term arose from the fact that in differential leveling in surveying we must construct horizontal line of sight, but due to collimation error the collimation line (i.e. line of sight) will not be 100% horizontal (by horizontal we mean tangent the level surface at the instrument position), instead it will be slightly deviated. so what we are looking to achieve when we eliminate the collimation error (using the 2 peg test) is a horizontal line of collimation.

A tacheometer is a type of theodolite used for rapid measurements. It determines the distance to the target electronically or electro-optically. Tacheometry is a system of rapid surveying that does not use chain or tape. There are times when measuring with a theodolite and measuring tape can be dangerous or impossible.A theodolite is an older surveying tool. Those were just precision optical devices used to compute angles between known fixed points, and they required using trigonometry to get the final values. However, there are modern versions of theodolites called Total Stations that are more convenient to use than even a tacheometer. These will cost you about $50,000 a piece or so.Please note that the question asks about a tacheometer (or tachymeter) which is a surveying tool, not a tachometer which measures engine speed.

The first proportional compasses were invented in the mid sixteenth century. Galileo invented a geometric and military compass in Padua in 1597. With his compass it was possible to make all sorts of arithmetical and geometrical calculations, from calculating interest to square and cube roots to drawing polygons and surveying.

The sine and cosine were originally developed for use in surveying. They provided a way to measure the distance across lakes and around mountains. Soon they were found to be useful in navigation. The sine was used to calculate pi. When electrical measurements were made, the sine law was used. If you want to know when to use the sine and when to use the cosine, you will need to get a trig book, a physics book, an astronomy book, a sailing book, and a few other books and read them all.

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That is called surveying.

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