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To think any civilization could build structures like the Babylonians did and not have a concept of angles is patently absurd. From what architecture they have

To think any civilization could build structures like the Babylonians did and not have a concept of angles is patently absurd. From what architecture they have that still stands, there is plenty of visual evidence that they indeed calculated for angles in some fashion.

jaclaz 9y ago
Yes and no. There is the concept of gradient (or slope). Even today you will find that earthworks (of course it depends on coutry regulations) use normally fractions expressing the gradient. You dig (in normal terrain) sections with a 1/1 slope, depending on consistency, harder soil can be excavated with 1/2, 1/3, 1/4 or 1/5 (w/h). You normally create landfillings /for roads, railways, etc.) with 3/2 slopes, which is a very good example how something that is extremely complex expressed in angles, like arctan 2/3, can easily be represented in practice. As well, acqueducts or sewages, railways and roads are usually calculated longitudinally basing on percent (or per thousand) gradient, like 0.1%, 0.5% (h/l). Vertical walls, like the walls built to contain the terrain on the "mountain side" of roads create - if perfectly vertical - the optical effect of appearing "leaning" towards the road, and usually are built with a 5% to 10% "opening" gradient from bottom to top to avoid that effect (and to save concrete), without anyone actually ever calculating the angle at the base.
jacobolus 9y ago
They clearly had a concept of similar triangles, and a concept similar to our notion of “slope” or “tangent” (apparently they actually used the reciprocal of slope, i.e. length per unit depth). However, they may not have had the concept as of ~4000 years ago of “angle measure” per se. You’d have to ask an expert in the history of astronomy when that concept was first developed semi-formally; my understanding is that it was 1000–1500 years later by subsequent Mesopotamian astronomers. Angle measure (another word for circular arclength corresponding to a particular rotation) is actually quite a tricky thing to get right. It’s the logarithm of the rotation (a pure bivector quantity) with the orientation stripped away – if we call our rotation R oriented in the plane of the unit bivector i, it’s θ = log(R)/i. To go from an angle measure back to a “complex number” (scalar + bivector) with which you can do anything useful, you need to put the orientation back in and then exponentiate it: R = exp(iθ). In dimensions higher than 2, you need to use rotation operators via a kind of sandwich product, because composition of rotations is not commutative. This is not an obvious concept, didn’t really get figured out properly until the late 19th century, and still isn’t taught very clearly to students today.

jaclaz 9y ago

Yes and no. There is the concept of gradient (or slope). Even today you will find that earthworks (of course it depends on coutry regulations) use normally fractions expressing the gradient. You dig (in normal terrain) sections with a 1/1 slope, depending on consistency, harder soil can be excavated with 1/2, 1/3, 1/4 or 1/5 (w/h). You normally create landfillings /for roads, railways, etc.) with 3/2 slopes, which is a very good example how something that is extremely complex expressed in angles, like arctan 2/3, can easily be represented in practice. As well, acqueducts or sewages, railways and roads are usually calculated longitudinally basing on percent (or per thousand) gradient, like 0.1%, 0.5% (h/l). Vertical walls, like the walls built to contain the terrain on the "mountain side" of roads create - if perfectly vertical - the optical effect of appearing "leaning" towards the road, and usually are built with a 5% to 10% "opening" gradient from bottom to top to avoid that effect (and to save concrete), without anyone actually ever calculating the angle at the base.

jacobolus 9y ago

They clearly had a concept of similar triangles, and a concept similar to our notion of “slope” or “tangent” (apparently they actually used the reciprocal of slope, i.e. length per unit depth). However, they may not have had the concept as of ~4000 years ago of “angle measure” per se. You’d have to ask an expert in the history of astronomy when that concept was first developed semi-formally; my understanding is that it was 1000–1500 years later by subsequent Mesopotamian astronomers. Angle measure (another word for circular arclength corresponding to a particular rotation) is actually quite a tricky thing to get right. It’s the logarithm of the rotation (a pure bivector quantity) with the orientation stripped away – if we call our rotation R oriented in the plane of the unit bivector i, it’s θ = log(R)/i. To go from an angle measure back to a “complex number” (scalar + bivector) with which you can do anything useful, you need to put the orientation back in and then exponentiate it: R = exp(iθ). In dimensions higher than 2, you need to use rotation operators via a kind of sandwich product, because composition of rotations is not commutative. This is not an obvious concept, didn’t really get figured out properly until the late 19th century, and still isn’t taught very clearly to students today.