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Lorenzo Mascheroni 's 1797 work La geometria del compasso, which develops a geometry based solely on compass constructions, is considered by the author as stepping back behind the “demarcation line” of Euclidean geometry. In this work Mascheroni emphasizes the practical aspects of this geometry over a theoretical approach. A century later, in 1899, David Hilbert and his student Michael Feldblum proposed a totally different approach – algebraic and axiomatic – concerning geometric constructions based on various instruments. Taking into account that, at the end of the 18th century, straightedge geometry was also developed, one may ask what happened to the image of instrument-based geometry during the 19th century? By focusing on Mascheroni's book and its reception, this article aims to examine the various views and conceptions of mathematicians with respect to this geometry.

The accuracy of 5-axis machine tools is a key factor to modern manufacturing of multi-axes machined workpieces. Up to 75% of errors on manufactured workpieces are caused by thermally induced errors of machine tools. In this paper a thermal test piece for evaluating thermal errors of machine tools equipped with rotary tables is introduced. In comparison to geometric test pieces, which detect geometrical errors of machine tools only, the developed thermal test piece visualizes thermally induced errors in X-, Y- and Z-direction as well as two additional angular errors at the tool center point (TCP) of 5-axis machine tools over time. Further, the thermal material expansion error of the test piece on centric clamped machined workpieces can be evaluated with the new thermal test piece. The presented thermal test piece is machined during a corresponding eight hour test cycle and tested on two 5-axis milling machines with different axis configurations. The thermal test piece can either be evaluated with a coordinate measuring machine (CMM) or manually with handheld measurement devices such as dial gauge, micrometer and straightedge. The thermal behavior of the two machine tools is also investigated, following the ISO 230-3:2007 regarding thermal errors of machine tools, under no-load or finishing conditions using the R-Test measurement setup. The resulting workpiece errors evaluated by using the thermal test piece are compared to the machine tool measurements. A good correspondence between the R-Test measurements and the thermal workpiece errors can be shown. CMM and handheld measurements are both capable to detect the thermal machine tool errors.

It is a well-known fact that there exists an angle that cannot be trisected with a straightedge and a compass. In general, it is impossible to divide an arbitrary angle into n-angles equally with only a straightedge and a compass, where n is a positive integer. We give an efficient algorithm to divide an arbitrary angle into n-angles almost equally with only a straightedge and a compass. Using this method, we can construct an almost regular n-gon for arbitrary n.

This paper describes a three-probe system that can be used to measure the parallelism and straightness of a pair of rails simultaneously. The parallelism is measured using a modified reversal method, while the straightness is measured using a sequential two-point method. The measurement algorithms were analyzed numerically using a pair of functionally defined rails to validate the three-probe system. Tests were also performed on a pair of straightedge rails with a length of 250 mm and a maximum straightness deviation of 0.05 μm, as certified by the supplier. The experimental results demonstrated that the parallelism measurement algorithm had a cancellation effect on the probe stage motion error. They also confirmed that the proposed system could measure the slope of a pair of rails with an accuracy of about 0.06 μrad. Therefore, by combining this technique with a sequential differential method to measure the straightness of the rails simultaneously, the surface profiles can be determined accurately and the stage error eliminated. The measured straightness deviation of each straightedge was less than 0.05 μm, consistent with the certified value.

According to international standards measurement uncertainties have to be stated with all measurements, including results from machine tool testing. For average industrial conditions measurement uncertainties are estimated for positioning measurements with laser interferometers and linear scales, for straightness measurements with laser interferometer and straightedge, and for pitch measurements. The results, i.e. the test uncertainties, are discussed and compared with average tolerances given in ISO standards for the corresponding test procedures. Uncertainties are not estimated for optimum conditions in order to make more people aware, that the alignment of the measurement device, temperature effects and repeatability of the machine tool and the measurement device might cause larger uncertainties than the uncertainty of the device alone.

This paper describes an industrial comparison of coordinate measuring machines (CMMs) carried out in the Scandinavian countries from October 1994 to May 1996. Fifty-nine industrial companies with a total of 62 CMMs participated in the project and measured a comparison package with five items chosen to represent a variety of dimensions, angles, and other geometrical quantities. A tool holder, two gauge blocks, a straightedge, and a ring together with instructions on how to measure the items were produced and sent to each participant. Simple measurement tasks were observed to be carried out with good results for the majority of the participants; whereas, increasing the level of difficulty from simple length measurements to more complex geometrical quantities gave severe problems for some of the companies. This occurred even though the participants measured according to prescribed procedures. An important part of the intercomparison was to test the ability of the participants to determine measurement uncertainties. One of the uncertainties was based upon a “best guess” but nevertheless, many participants did not even report this uncertainty. Uncertainty budgeting was not used for measurements other than simple length. For each company, a comparison of their measurement ability with the reference laboratory and other Scandinavian companies was made possible. A network regarding CMMs was created in these Scandinavian countries.

A method is described to measure the straightness of travel of the carriage of a single-point diamond turning machine. The method also measures the slide parallelism to the workhead spindle. A cylindrical artifact was produced on the lathe and used as a straightedge. A flat land was cut in the artifact that was parallel to the workhead axis of rotation. The flat land was measured with a LVDT indicator. The method was demonstrated on a Moore Special Tool Co. M18 Aspheric Generator. The quality of measurements achieved were better than 0.1 μm.

An off-line Geometric Adaptive Control (GAC) scheme is proposed to compensate for machining straightness errors due to the machine tool's inaccuracy and those arising as a result of the metal cutting process during the finish peripheral end milling process. In the milling process, the workpiece travels along the guideway while the spindle system remains fixed. The scheme is based on the exponential smoothing of post-process measurements of relative machining errors due to the tool and bed deflections. Without a priori knowledge of the variations of the cutting parameters, the time-varying parameters are estimated by an Exponentially Weighted Recursive Least Squares (EWRLS) method. This method is able to incorporate a straightedge which is not necessarily accurate to identify the guideway errors. To reduce the drift of the cutting parameters, a single parameter adaptation method is introduced. Experimental results show that the location error is controlled within the range of the fixing error of the milling bed on the guideway. Further, the waviness error is reduced to less than 10 μm in the machining of a 508-mm long prismatic workpiece regardless of the machining conditions.

A feasibility study is made for the utility and the application of laser beam as a straight datum. By using the quadrant photodiode position sensor and the piezo-compensator with nanometer resolution, the carriage on a slide table is so designed and built as to travels straight along the laser beam as a “perfect” straight datum. Using this set up, the geometrical accuracy of an optical flat and a precision straightedge is measured by a displacement sensor fixed on the carriage in “perfect” straight motion with sufficient resolution and repeatability of 3 nm over the measuring distance of about 100 mm.

Mandibular denture teeth should be set in a manner to reduce lateral forces on the residual ridge. This is best accomplished by placing the central groove of the mandibular posterior tooth over the crest of the ridge. Usually, landmark lines are placed on the master cast and tooth position is verified with a straightedge. This article presents an alternative method to verify buccolingual tooth position.