• Journal of the Korean School Mathematics Society
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• v.11 no.1
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• pp.19-30
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• 2008

Cavalieri is chiefly remembered for his work on the problem "indivisibles." Building on the work of Archimedes, he investigated the method of construction by which areas and volumes of curved figures could be found. Cavalieri regarded an area as made up of an indefinite number of parallel line segments and a volume of an indefinite number of parallel plane areas. He called these elements the indivisibles of area and volume. Cavalieri developed a method of the indivisibles which he used to determine areas and volumes. We call this Cavalieri's principle which states that there exists a plane such that any plane parallel to it intersects equal areas In both objects, then the volumes of the two objects are equal. Cavalieri's principle and method of the indivisibles are very important to understand of volume of a pyramid for gifted students.

• Journal for History of Mathematics
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• v.21 no.2
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• pp.19-38
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• 2008

This study aims to investigate the evolution of calculating the volume of a sphere in eastern and western mathematical history. In western case, Archimedes', Cavalieri's and Kepler's approaches, and in eastern case, Nine Chapters';, Liu Hui's and Zus' approaches are worthy of noting. The common idea of most of these approaches is the infinitesimal concept corresponding to Cavalieri's or Liu-Zu's principle which would developed to the basic idea of Calculus. So this study proposes an alternative to organization of math-textbooks or instructional procedures for teaching the volume of a sphere based on the principle.

• Proceedings of the Optical Society of Korea Conference
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• 2006.07a
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• pp.373-374
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• 2006

• Journal of Periodontal and Implant Science
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• v.49 no.1
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• pp.58-58
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• 2019

• School Mathematics
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• v.8 no.1
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• pp.27-43
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• 2006

The objective of this paper is to design teaching units to foster secondary pre-service teachers' mathematising abilities. In these teaching units we focus on generalizing area of a 2-dimensional triangle and volume of a 3-dimensional tetrahedron to n-volume of n-simplex In this process of generalizing, principle of the permanence of equivalent forms and Cavalieri's principle are applied. To find n-volume of n-simplex, we define n-orthogonal triangular prism, and inquire into n-volume of it. And we find n-volume of n-simplex by using vectors and determinants. Through these teaching units, secondary pre-service teachers can understand and inquire into n-simplex which is generalized from a triangle and a tetrahedron, and n-volume of n-simplex which is generalized from area of a triangle and volume of a tetrahedron. They can also promote natural connection between school mathematics and academic mathematics.

• Journal of Korean Neurosurgical Society
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• v.58 no.5
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• pp.454-461
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• 2015

Objective : In this study, we aimed to investigate the underlying ethiological factors in chiari malformation (CM) type-I (CMI) via performing volumetric and morphometric length-angle measurements. Methods : A total of 66 individuals [33 patients (20-65 years) with CMI and 33 control subjects] were included in this study. In sagittal MR images, tonsillar herniation length and concurrent anomalies were evaluated. Supratentorial, infratentorial, and total intracranial volumes were measured using Cavalieri method. Various cranial distances and angles were used to evaluate the platybasia and posterior cranial fossa (PCF) development. Results : Tonsillar herniation length was measured $9.09{\pm}3.39mm$ below foramen magnum in CM group. Tonsillar herniation/concurrent syringomyelia, concavity/defect of clivus, herniation of bulbus and fourth ventricle, basilar invagination and craniovertebral junction abnormality rates were 30.3, 27, 18, 2, 3, and 3 percent, respectively. Absence of cisterna magna was encountered in 87.9% of the patients. Total, IT and ST volumes and distance between Chamberlain line and tip of dens axis, Klaus index, clivus length, distance between internal occipital protuberance and opisthion were significantly decreased in patient group. Also in patient group, it was found that Welcher basal angle/Boogard angle increased and tentorial slope angle decreased. Conclusion : Mean cranial volume and length-angle measurement values significantly decreased and there was a congenital abnormality association in nearly 81.5 percent of the CM cases. As a result, it was concluded that CM ethiology can be attributed to multifactorial causes. Moreover, congenital defects can also give rise to this condition.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1001-1015
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• 2010

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

• Journal of Elementary Mathematics Education in Korea
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• v.13 no.1
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• pp.115-140
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• 2009

Freudenthal has advocated the mathematisation theory. Mathematisation is an activity which endow the reality with order, through organizing phenomena. According to mathematisation theory, the departure of children's learning of mathematics is not ready-made formal mathematics, but reality which contains mathematical germination. In the first place, children mathematise reality through informal method, secondly this resulting reality is mathematised by new tool. Through survey, it turns out that area unit of Korea's seventh elementary mathematics textbook is not correspond to mathematisation theory. In that textbook, the area formular is hastily presented without sufficient real context, and the relational understanding of area concept is overwhelmed by the practice of the area formular. In this thesis, first of all, I will reconstruct area unit of seventh elementary textbook according to Freudenthal's mathematisation theory. Next, I will perform teaching experiment which is ruled by new lesson design. Lastly, I analysed the effects of teaching experiment. Through this study, I obtained the following results and suggestions. First, the mathematisation was effective on the understanding of area concept. Secondly, in both experimental and comparative class, rich-insight children more successfully achieved than poor-insight ones in the task which asked testee comparison of area from a view of number of unit square. This result show the importance of insight in mathematics education. Thirdly, in the task which asked testee computing area of figures given on lattice, experimental class handled more diverse informal strategy than comparative class. Fourthly, both experimental and comparative class showed low achievement in the task which asked testee computing area of figures by the use of Cavalieri's principle. Fifthly, Experiment class successfully achieved in the area computing task which resulting value was fraction or decimal fraction. Presently, Korea's seventh elementary mathematics textbook is excluding the area computing task which resulting value is fraction or decimal fraction. By the aid of this research, I suggest that we might progressively consider the introduction that case. Sixthly, both experimental and comparative class easily understood the relation between area and perimeter of plane figures. This result show that area and perimeter concept are integratively lessoned.

• Journal for History of Mathematics
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• v.27 no.2
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• pp.127-138
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• 2014

In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

• Clinical and Experimental Reproductive Medicine
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• v.40 no.3
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• pp.115-121
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• 2013

Objective: Letrozole and estradiol valerate are used to treat some hormonally-responsive symptoms and also in modeling of the polycystic ovary syndrome. However, the stereological analysis of the ovary has received less attention. Estimation of the whole ovary volume using the Cavalieri method can be applied in any orientation desired, but estimation of the mean volume of the oocytes requires isotropic uniform random sectioning. Here, a combined method was developed for estimating the parameters. To our knowledge, no comparison has been made of the effects of letrozole and estradiol on the ovary. Methods: Sixty rats were divided into 4 groups receiving estradiol (4 mg/kg), olive oil, letrozole (1 mg/kg), or normal saline. After 21 days, their ovaries were studied. Results: Relative to the control group, the total volume of the ovary and the cortex increased in the letrozole-treated and estradiol-treated rats. In addition, the number of the preantral, antral, and granulosa cells decreased by 43% to 56% in the letrozole- and estradiol-treated rats. On average, a 19% increase was observed in the atretic oocytes of the letrozole-treated and estradiol-treated rats, but the mean oocyte volume decreased by 29% to 44% in letrozole- and estradiol-treated rats. Furthermore, the letrozole-treated rats showed a 5-fold and 7-fold increase in the volume of the cysts and corpus luteum, respectively. A 3-fold increase was found in the volume of both the cysts and corpus luteum in the estradiol group. Conclusion: The structural changes of the ovary were most pronounced in the letrozole-treated animals.