How did Roman civil engineers calculate angles? Did they use Greek or Arabic notation? Or Babylonian?

You don't have to count to 45 to use a 45 degree angle. Most of the basic angles are easily constructed with a straight edge and a compass. A circle only has 360 degrees because it is easily divisible (2,3,4,5,6,8,9,10, etc..) Think more of the way an inch is divided, into halves.

The word construction in geometry has a very specific meaning: the drawing of geometric items such as lines and circles using only a compass and straightedge. Very importantly, you are not allowed to measure angles with a protractor, or measure lengths with a ruler. The ancient Greek mathematician Euclid is the acknowledged inventor of geometry (the word geometry itself derives from the Greek γαία (gaía), earth, land + μετρέω (metréo) to measure, meaning: to measure, to survey land). He did this over 2000 years ago, and his book "Elements" is still regarded as the ultimate geometry reference. In that work, he uses these construction techniques extensively, and so they have become a part of the geometry field of study. Why didn't Euclid just measure things with a ruler and calculate lengths? For example, one of the basic constructions is bisecting a line (dividing it into two equal parts). Why not just measure it with a ruler and divide by two? The answer is surprising. The Greeks could not do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number - the only kind they had. Also, their numbers did not use a positional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In short, they could perform very little useful arithmetic. For more info and some practical examples follow the link below:

Roman masonry and timber buildings were constructed with physical tools. Although the ancient Romans are generally famous for their advanced engineering accomplishments, most of their own inventions were improvements on older ideas, concepts and inventions. Cement was originally invented in Egypt, although the Romans improved the formula. Technology for bringing running water into cities was also invented in the east. The architecture used in Rome was strongly influenced by Greek and Etruscan sources. Roads were common at that time, but the Romans improved their design and perfected the construction methodologies to the extent that many of their roads are still in use today. The main angle of the Roman empire was 90%. It was a culture that developed out of a military need. Indeed, military engineering was in many ways institutionally endemic in Roman military culture, as demonstrated by the fact that each Roman legionary had as part of his equipment a shovel, alongside his gladius (sword) and pila (spears). Each Roman legion had a military legionary fort as its permanent base. However, when on the march, particularly in enemy territory, the legion would, after a day's marching, construct a fortified camp or castra, requiring as raw materials only earth, turf and timber. Camp construction were the responsibility of special engineering units to which specialists of many types belonged, officered by architecti (engineers), from a class of troops known as immunes since they were excused from or, literally, immune from, regular duties. These engineers would requisition manual labor from the soldiers at large as required. A legion could throw up a camp under enemy attack in as little as a few hours. Judging from the names, they probably used a repertory of camp plans from a set textbook, selecting the one appropriate to the length of time a legion would spend in it: tertia castra, quarta castra: "a camp of three days", "four days", etc. The Roman army also took part in building projects for civilian use. There was sound reasons for the use of the army in building projects: primarily, that if they weren't directly engaged in military campaigns, the legions were largely unproductive, costing the Roman state large sums of money. But the involvement of the soldiers in building works, kept them not only well accustomed to hard physical labour, but also kept them busy, since it was the widely held belief that busy armies weren't plotting to mutiny, whereas idle armies were. For gradient, and the constructiuon of aqueducts, they used a chorobates. The chorobates was used to level terrain before construction. It was a wooden object supported by four legs with a flat board on top in which was engraved a half circle. When used the half circle was filled with water and the angle at which there was no water was measured. For grids, they used two main devices, the rod and another called the groma, which helped them obtain right angles. Gromas were used to measure right angles. A groma was the singular most important instrument of the mensores, or Roman military surveyors, and their commander the praefectus castrorum. The groma was used to mark-out the two main axes of the camp; the cardo decumanus which pointed in the direction from which the main danger would arise, and the cardo maximus which lay at right-angles to it. The groma consisted of a cruciform metal frame, set at meticulous right angles, with plumb-lines suspended from the extremities of each arm. The instrument was suspended from a tall staff set slightly off-centre, which would allow the surveyor to make sightings along the plumb-lines unimpeded by the pole itself. The gromatici, the Roman equivalent of rod men, placed rods and put down a line called the rigor. As they did not possess anything like a transit, an architect tried to achieve straightness by looking along the rods and commanding the gromatici to move them as required. Using the gromae they then laid out a grid pattern. The main axes of the camp would then be measured by the mensores - using accurately cut poles perhaps ten feet in length - and marked-out using coloured flags, along the line indicated by the groma. The Groma was also the name of the focus of the camp - the point where the two main streets crossed before the entrance to the headquarters building - where the like-named surveying instrument was initially placed. Once they had a square marked out, they could acheive 45% by dividing this along the diagional. for 30% or 60% they built a simple 3-4-5 tiangle. Circles were constructed with rope, and Elipses with two fulcrums. For more demanding jobs they developed another instrument, an goniometer It looked like dish mounted on to of a table, with two veiw sites fited to a pivot. It enabled them to work out accurate angles in topography, civil engineering work or even in architecture. Two plum lines allow the user to ensure the instrument is horizontal. Alternatively a water container could be used as a spirit level. Looking through the aiming site would allow you to point the instrument and measure angles versus a second landmark. The romans had aquired the Babalonian maths from the Greeks, thus divided the world into 360 degrees, but were still working from base ten - not 12. It was not until the christians arrived that 12 was used in length measurements. Oh by the way Damian, the Greeks could divide 5 by 2 and they got 2 1/2, they used fractions. 22/7 is pye, another greek invention long before the indian point was added to make it 3.14

Well theSumerians/Assyrians/Babylonians used Geometry before the Greeks. People often forget that the Greeks learned much of what they knew from much older civilisations. All these peoples were before the romans, so they probably took the knowledge from all of them. NOTE: Arab Civilisation did not exist in the time of the Romans! So it definately was not them. Arab civilisation started in 7th Century AD, prior to that they mostly lived as tribes in what is now Saudi Arabia.

They would probably have used Greek for their calculations, because geometry came from the Greeks.