Cavalieri's Principle


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Cavalieri’s Principle

 

If the areas of the cross sectionsof two solids by any plane parallel to a given plane are invariablyequal, then the two solids have the same volume. This proposi-tion (and the analogous one for plane figures), which was alreadywell-known to ancient Greek mathematicians, is usually calledCavalieri’s principle, although the Italian mathematicianF. B. Cavalieri in his Geometry (1635) does not take it as aprinciple but proves it.

References in periodicals archive ?
Whereas the stereology based on Cavalieri's principle used mostly in vivo studies and in MR images is an unbiased method that may estimate the volume of an object divided into a sequence of parallel cross sectional planes (Gundersen et al., 1988; Roberts et al., 2000).
In this study, we calculated the volumes of maxillary sinuses on computed tomographic (CT) scans of paranasal sinuses using the Cavalieri's principle to explore whether nasal septal deviations that deflect the hard palate affect the volume of the maxillary sinus.
Cavalieri's principle was used for calculating the total volumes.
We used the ImageJ software with the planimetry method and Cavalieri's principle in our calculations.
The Cavalieri's principle can give some inspiration.
The idea is based on Cavalieri's principle. In software Cabri 3D we realize first experiments.
This circumstantial evidence indicates application of Cavalieri's principle. If the areas of plane sections are equal then the volumes are equal, too.
By Cavalieri's Principle (Cavalieris_Principle.cg3) this implies that the volumes of the bodies are equal and hold
The total volume of white matter, V(wm), was calculated from Cavalieri's principle (Gundersen et al., 1988; Tang and Nyengaard, 1997; Tang et al., 1997; Marner et al., 2003; Tang et al., 2003; Tang and Nyengaard, 2004):
Thus, we evaluated the impact of VCT with the Vasotrain-447 on diabetic foot ulcers using the stereological method based on Cavalieri's principle. We hypothesized that the diabetic foot ulcer surface area would decrease from the vacuum-compression effects of the Vasotrain-447.
In this research, we estimated before and after treatment the foot ulcer surface area stereologically using the regular point-counting grid method based on Cavalieri's principle [14-15].