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克莱因瓶是一个不可定向的二维紧流形，而球面或轮胎面是可 克莱因瓶 克莱因瓶 定向的二维紧流形。如果观察克莱因瓶，有一点似乎令人困惑－－克莱因瓶的瓶颈和瓶身是相交的，换句话说，瓶颈上的某些点和瓶壁上的某些点占据了三维空间中的同一个位置。我们可以把克莱因瓶放在四维空间中理解：克莱因瓶是一个在四维空间中才可能真正表现出来的曲面。如果我们一定要把它表现在我们生活的三维空间中，我们只好将就点，把它表现得似乎是自己和自己相交一样。克莱因瓶的瓶颈是穿过了第四维空间再和瓶底圈连起来的，并不穿过瓶壁。用扭结来打比方，如果把它看作平面上的曲线的话，那么它似乎自身相交，再一看似乎又断成了三截。但其实很容易明白，这个图形其实是三维空间中的曲线。它并不和自己相交，而是连续不断的一条曲线。在平面上一条曲线自然做不到这样，但是如果有第三维的话，它就可以穿过第三维来避开和自己相交。只是因为我们要把它画在二维平面上时，只好将就一点，把它画成相交或者断裂了的样子。克莱因瓶也一样，我们可以把它理解成处于四维空间中的曲面。在我们这个三维空间中，即使是最高明的能工巧匠，也不得不把它做成自身相交的模样；就好像最高明的画家，在纸上画扭结的时候也不得不把它们画成自身相交的模样。有趣的是，如果把克莱因瓶沿着它的对称线切下去，竟会得到两个莫比乌斯环。在二维看似穿过自身的绳子 在二维看似穿过自身的绳子 如果莫比乌斯带能够完美的展现一个“二维空间中一维可无限扩展之空间模型”的话，克莱因瓶只能作为展现一个“三维空间中二维可无限扩展之空间模型”的参考。因为在制作莫比乌斯带的过程中，我们要对纸带进行180°翻转再首尾相连，这就是一个三维空间下的操作。理想的“三维空间中二维可无限扩展之空间模型”应该是在二维面中，朝任意方向前进都可以回到原点的模型，而克莱因瓶虽然在二维面上可以向任意方向无限前进。但是只有在两个特定的方向上才会回到原点，并且只有在其中一个方向上，回到原点之前会经过一个“逆向原点”，真正理想的“三维空间中二维可无限扩展之空间模型”也应该是在二维面上朝任何方向前进，都会先经过一次“逆向原点”，再回到原点。而制作这个模型，则需要在四维空间上对三维模型进行扭曲。数学中有一个重要分支叫“拓扑学”，主要是研究几何图形连续改变形状时的一些特征和规律的，克莱因瓶和莫比乌斯带变成了拓扑学中最有趣的问题之一。莫比乌斯带的概念被广泛地应用到了建筑，艺术，工业生产中。三维空间里的克莱因瓶 拓扑学的定义编辑 克莱因瓶定义为正方形区域 [0,1]×[0,1] 模掉等价关系(0,y)~(1,y), 0≤y≤1 和 (x,0)~(1-x,1), 0≤x≤1。类似于 Mobius Band, 克莱因瓶不可定向。但 Mobius 带可嵌入   ，而克莱因瓶只能嵌入四维（或更高维）空间。莫比乌斯带编辑 把一条纸带的一段扭180°，再和另一端粘起来就得到一条莫比乌斯带的模型。这也是一个只有莫比乌斯带、一个面的曲面，但是和球面、轮胎面和克莱因瓶不同的是，它有边（注意，它只有一条边）。如果我们把两条莫比乌斯带沿着它们唯一的边粘合起来，你就得到了一个克莱因瓶 莫比乌斯带 莫比乌斯带 （当然不要忘了，我们必须在四维空间中才能真正有可能完成这个粘合，否则的话就不得不把纸撕破一点）。同样地，如果把一个克莱因瓶适当地剪开来，我们就能得到两条莫比乌斯带。除了我们上面看到的克莱因瓶的模样，还有一种不太为人所知的“8字形”克莱因瓶。它看起来和上面的曲面完全不同，但是在四维空间中它们其实就是同一个曲面－－克莱因瓶。实际上，可以说克莱因瓶是一个3°的莫比乌斯带。我们知道，在平面上画一个圆，再在圆内放一样东西，假如在二度空间中将它拿出来，就不得不越过圆周。但在三度空间中，很容易不越过圆周就将其拿出来，放到圆外。将物体的轨迹连同原来的圆投影到二度空间中，就是一个“二维克莱因瓶”，即莫比乌斯带（这里的莫比乌斯带是指拓扑意义上的莫比乌斯带）。再设想一下，在我们的3°空间中，不可能在不打破蛋壳的前提下从鸡蛋中取出蛋黄，但在四度空间里却可以。将蛋黄的轨迹连同蛋壳投影在三度空间中，必然可以看到一个克莱因瓶。制造经历编辑 过去，德国数学家克莱因就曾提出了“不可能”设想，即拓扑学的大怪物－－克莱因瓶。这种瓶子根本没有内、外之分，无论从什么地方穿透曲面，到达之处依然在瓶的外面，所以，它本质上就是一个“有外无内”的古怪东西。尽管现代玻璃工业已经发展得非常先进，但是，所谓的“克莱因瓶”却始终是大数学家克莱因先生脑子里头的“虚构物”，根本制造不出来。许多国家的数学家老是想造它一个出来，作为献给国际数学家大会的礼物。然而，等待他们的是一个失败接着一个失败。也有人认为，即使造不出玻璃制品，能造出一个纸模型也不错。如果真的解决了这个问题，那可是个大收获！直径和年龄 最新的研究认为宇宙的直径可920亿光年，甚至更大。[28] 目前可观测的宇宙年龄大约为138.2亿年。[29] 形状 宇宙微波背景的温度一端高，暗示呈弯曲状 宇宙微波背景的温度一端高，暗示呈弯曲状 [30] 目前的宇宙理论认为宇宙可能是类似马鞍状的负弯曲形状，该理论源于宇宙大爆炸理论，整个宇宙的外形如同一个吹起的气球，我们则生活在宇宙的“表面”。[31] 同时，科学家也认为宇宙是平坦的，根据美国宇航局的调查，宇宙可能是平坦的，2013年的调查发现如果宇宙是平坦的，那么误差只有0.4%。[32] 斯蒂芬·霍金表示，我们宇宙的形状可能是一种难以置信的几何图形，更接近于超现实主义的艺术，如同荷兰艺术家摩里茨·科奈里斯·埃舍尔创 银河系 银河系 [33] 作的图形一样。霍金的想法以弦理论为依据，而该理论目前仍然还处于假设之中，并未被验证。如果用语言来形容宇宙的形状，应该是整体呈现多重镶嵌模式，具有无限重复出现的扭曲面，曲面间环环相扣，如同科奈里斯·埃舍尔创作的“圆形极限IV”图案，也与美国工程师P.H. Smith创作的“史密斯圆图”类似，体现出双曲空间的概念，是一种非欧几何的空间形态。[34] 层次结构 当代天文学研究成果表明，宇宙是有层次结构的、 即将发生碰撞的两个星系NGC 470和NGC 474 即将发生碰撞的两个星系NGC 470和NGC 474 [35] 不断膨胀、物质形态多样的、不断运动发展的天体系统。行星、小行星、彗星和流星体都围绕中心天体太阳运转，构成太阳系。太阳系外也存在其他行星系统。约2500亿颗类似太阳的恒星和星际物质构成更巨大的天体系统——银河系。银河系的直径约10万光年，太阳位于银河系的一个旋臂中，距银心约2.6万光年。银河系外还有许多类似的天体系统，称为河外星系，常简称星系。目前观测到1000亿个星系，科学家估计宇宙中至少有2万亿个星系。星系聚集成大大小小的集团，叫星系团。平均而言，每个星系团约有百余个星系，直径达上千万光年。现已发现上万个星系团。包括银河系在内约40个星系构成的一个小星系团叫本星系群。椭圆星系Hercules A中心超大黑洞引发的喷流 椭圆星系Hercules A中心超大黑洞引发的喷流 [36] 若干星系团集聚在一起构成的更高一层次的天体系统叫超星系团。超星系团往往具有扁长的外形，其长径可达数亿光年。通常超星系团内只含有几个星系团，只有少数超星系团拥有几十个星系团。本星系群和其附近的约50个星系团构成的超星系团叫做本超星系团。星系分类 根据可反映星系发展状态的序列号对星系进行了分类，可以粗略地将星系划分出椭圆星系、透镜星系、漩涡星系、棒旋星系和不规则星系等五种。[37] 太阳系天体 太阳质量占太阳系总质量的99.86%，它以自己强大的引力将 NASA公布的太阳风暴的照片 NASA公布的太阳风暴的照片 [38] 太阳系里的所有天体牢牢地吸引在它的周围，使它们不离不散、井然有序地绕自己旋转。同时，太阳又作为一颗普通恒星，带领它的成员，万古不息地绕银河系的中心运动。[39]  太阳的半径为696000千米，质量为1.989×10^30kg，中心温度约15000000 ℃，。[40]  如果一个人站在太阳表面，那么他的体重将会是在地球上的20倍。[41]  现代星云假说根据观测资料和理论计算，提出：太阳系原始星云是巨大的星际云瓦解的一个小云，一开始就在自转，并在自身引力作用下收缩，中心部分形成太阳，外部演化成星云盘，星云盘以后形成行星。目前，现代星云说又存在不同学派，这些学派之间还存在着许多差别，有待进一步研究和证实。[42] 金星是离太阳的第二颗行星，夜空中亮度仅次于月球。[43]  金星上没有水，大气中严重缺氧，二氧化碳占97%以上，空气中有一层厚达20千米至30千米的浓硫酸云，地面温度从不低于400℃，是个名副其实的“炼狱”般世界。金星地面的大气压强为地球的90倍，相当于地球海洋中900米深度时的压强。金星大气主要由二氧化碳等温室气体组成，失控的温室效应，是导致金星极端气候的主要原因。由于金星没有内禀磁层保护，诱发磁层中磁场重联释放的巨大能量，使得金星大气被加热后加速逃逸。科学界认为，金星上大气的逃逸，是造成金星上缺水而被富含二氧化碳的稠密大气所笼罩，从而导致严重的温室效应的原因。[44] 木星是离太阳第五颗行星，而且是最大的一颗，比所有其他的行星 木星及其卫星欧罗巴（木卫二） 木星及其卫星欧罗巴（木卫二） [45] 的合质量大2倍（地球的318倍），直径142987km。它是气态行星没有实体表面，由90%的氢和10%的氦（原子数之比, 75/25%的质量比）及微量的甲烷、水、氨水和“石头”组成。这与形成整个太阳系的原始的太阳系星云的组成十分相似。木星可能有一个石质的内核，相当于10－15个地球的质量。内核上则是大部分的行星物质集结地，以液态氢的形式存在。液态金属氢由离子化的质子与电子组成（类似于太阳的内部，不过温度低多了）。木星共有67颗木卫。按距离木星中心由近及远的次序为：木卫十六、木卫十四、木卫五、木卫十五、木卫一、木卫二、木卫三、木卫四、木卫十三、木卫六、木卫十、木卫七、木卫十二、木卫十一、木卫八和木卫九。[46] 水星是最接近太阳的行星。水星的半径约为2440公里，在八大行星中是最小的。水星昼夜温差极大，白天摄氏 430 度，晚上约可达零下170 度，是太阳系八大行星中温差最大的一个行星。[47]  水星的外大气层非常稀薄，是由水星表面和太阳风中的原子和离子构成。[48]  科学家确认水星表面含有丰富的碳，认为碳是水星表面呈黑色的原因，水星表面的岩石是由低重量百分比的石墨碳构成。[49] “好奇号”火星探测器在火星表面采集样本 “好奇号”火星探测器在火星表面采集样本 [50] 火星是地球的近邻，是太阳系由内往外数第四颗行星。直径6794km，体积为地球的15%，质量为地球的11%。火星表面是一个荒凉的世界，空气中二氧化碳占了95%。火星大气十分稀薄，密度还不到地球大气的1%，因而根本无法保存热量。这导致火星表面温度极低，很少超过0℃，在夜晚，最低温度则可达到-123℃。火星被称为红色的行星，这是因为它表面布满了氧化物，因而呈现出铁锈红色。其表面的大部分地区都是含有大量的红色氧化物的大沙漠，还有赭色的砾石地和凝固的熔岩流。火星上常常有猛烈的大风，大风扬起沙尘能形成可以覆盖火星全球的特大型沙尘暴。每次沙尘暴可持续数个星期。火星两极的冰冠和火星大气中含有水份。从火星表面获得的探测数据证明，在远古时期，火星曾经有过液态的水，而且水量特别大。[51] 土星是离太阳第六颗行星，直径120536㎞，体积仅次于木星。主要由氢组成，还有少量的氦与微量元素，内部的核心包括岩石和冰，外围由数层金属氢和气体包裹着。地球距离土星13亿公里。土星的引力比地球强2.5倍，能够牵引太阳系内其它行星，使地球处于一个椭圆轨道中运行，并且与太阳保持适当距离，适宜生命繁衍。当土星轨道倾斜20度将使地球轨道比金星轨道更接近太阳，同时，这将导致火星完全离开太阳系。[52]  土星是已知唯一密度小于水的行星，假如能够将土星放入一个巨大的浴池之中，它将可以漂浮起来。土星有一个巨大的磁气圈和一个狂风肆虐的大气层，赤道附近的风速可达1800千米/时。在环绕土星运行的31颗卫星中间，土卫六是最大的一颗，比水星和月球还大，也是太阳系中唯一拥有浓厚大气层的卫星。[53] 天王星是离太阳第七颗行星，51118km。体积约为地球的65倍，在九大行星中仅次于木星和土星。天王星的大气层中83%是氢，15%为氦，2%为甲烷以及少量的乙炔和碳氢化合物。上层大气层的甲烷吸收红光，使天王星呈现蓝绿色。大气在固定纬度集结成云层，类似于木星和土星在纬线上鲜艳的条状色带。天王星云层的平均温度为零下193摄氏度。质量为8.6810±13×10²⁵kg，相当于地球质量的14.63倍。密度较小，只有1.24克/立方厘米，为海王星密度值的74.7%。[54] 恒星 恒星 海王星是离太阳的第八颗行星，直径49532千米。海王星绕太阳运转的轨道半径为45亿千米，公转一周需要165年。海王星的直径和天王星类似，质量比天王星略大一些。海王星和天王星的主要大气成分都是氢和氦，内部结构也极为相近，所以说海王星与天王星是一对孪生兄弟。[55]  海王星有太阳系最强烈的风，测量到的时速高达2100公里。海王星云顶的温度是－218 °C，是太阳系最冷的地区之一。海王星核心的温度约为7000 °C，可以和太阳的表面比较。海王星在1846年9月23日被发现，是唯一利用数学预测而非有计划的观测发现的行星。[56] 冥王星，位于海王星以外的柯伊伯带内侧，是柯伊伯带中已知的最大天体。[57]  直径约为2370±20km，是地球直径的18.5%。[58]  2006年8月24日，国际天文学联合会大会24日投票决定，不再将传统九大行星之一的冥王星视为行星，而将其列入“矮行星”。大会通过的决议规定，“行星”指的是围绕太阳运转、自身引力足以克服其刚体力而使天体呈圆球状、能够清除其轨道附近其他物体的天体。在太阳系传统的“九大行星”中，只有水星、金星、地球、火星、木星、土星、天王星和海王星符合这些要求。冥王星由于其轨道与海王星的轨道相交，不符合新的行星定义，因此被自动降级为“矮行星”。[59]  冥王星的表面温度大概在-238到-228℃之间。冥王星的成份由70%岩石和30%冰水混合而成的。地表上光亮的部分可能覆盖着一些固体氮以及少量 卫星拍月球经过地球，可见清晰月球背面 卫星拍月球经过地球，可见清晰月球背面 [60] 的固体甲烷和一氧化碳，冥王星表面的黑暗部分可能是一些基本的有机物质或是由宇宙射线引发的光化学反应。冥王星的大气层主要由氮和少量的一氧化碳及甲烷组成。大气极其稀薄，地面压强只有少量微帕。[61] 地球是离太阳第三颗行星，是我们人类的家乡，尽管地球是太阳系中一颗普通的行星，但它在许多方面都是独一无二的。比如，它是太阳系中唯一一颗面积大部分被水覆盖的行星，也是目前所知唯一一颗有生命存在的星球。质量M=5.9742 ×10^24 公斤，表面温度：t = - 30 ～ +45。[62]  英国科研人员在《天体生物学》杂志上报告说，如果没有小行星撞击等可能剧烈改变环境的事件发生，地球适宜人类居住的时间还剩约17.5亿年，不过人为造成的气候变化可能缩短这一时间。[63] 彗星是由灰尘和冰块组成的太阳系中的一类小天体，绕日运动。[64]  科学家使用探测器对彗星的化学遗留物进行分析，发现其主要成份为氨、甲烷、硫化氢、氰化氢和甲醛。科学家得出结论称，彗星的气味闻起来像是臭鸡蛋、马尿、酒精和苦杏仁的气味综合。[65-66] “67P/楚留莫夫-格拉希门克”彗星 “67P/楚留莫夫-格拉希门克”彗星 [67] 在太阳系的周围还包裹着一个庞大的“奥尔特云”。星云内分布着不计其数的冰块、雪团和碎石。其中的某些会受太阳引力影响飞入内太阳系，这就是彗星。这些冰块、雪团和碎石进入太阳系内部，其表面因受太阳风的吹拂而开始挥发。所以彗星都拖着一条长长的尾巴，而且越靠近太阳尾巴越长、越明显。太阳系内的星际空间并不是真空的，而是充满了各种粒子、射线、气体和尘埃。[68] 柯伊伯带，是一种理论推测认为短周期彗星是来自离太阳50—500天文单位的一个环带，位于太阳系的尽头。柯伊伯带是冰质残片组成的巨环，位于海王星轨道之外，环绕着太阳系的外边缘。[69] 物质多样性 红巨星，当一颗恒星度过它漫长的青壮年期——主序星阶段，步入老年期时，它将首先变为一颗红巨星。称它为“巨星”，是突出它的体积巨大。在巨星阶段，恒星的体积将膨胀到十亿倍之多。称它为“红”巨星，是因为在这恒星迅速膨胀的同时，它的外表面离中心越来越远，所以温度将随之而降低，发出的光也就越来越偏红。不过，虽然温度降低了一些，可红巨星的体积是如此之大，它的光度也变得很大，极为明亮。红巨星一旦形成，就朝恒星的下一阶段白矮星进发。[70] 白矮星，是一种低光度、高密度、高温度的恒星。因为颜色呈白色、体积比较矮小，因此被命名为白矮星。哈勃望远镜观测到白矮星死亡过程 哈勃望远镜观测到白矮星死亡过程 [71] 白矮星是一种很特殊的天体，它的体积小、亮度低，但质量大、密度极高。白矮星是中低质量的恒星的演化路线的终点。在红巨星阶段的末期，恒星的中心会因为温度、压力不足或者核聚变达到铁阶段而停止产生能量。恒星外壳的重力会压缩恒星产生一个高密度的天体。一个典型的稳定独立白矮星具有大约半个太阳质量，比地球略大。这种密度仅次于中子星和夸克星。如果白矮星的质量超过1.4倍太阳质量，那么原子核之间的电荷斥力不足以对抗重力，电子会被压入原子核而形成中子星。原子是由原子核和电子组成的，原子的质量绝大部分集中在原子核上，在巨大的压力之下，电子将脱离原子核，成自由电子。这种自由电子气体将尽可能地占据原子核之间的空隙，从而使单位空间内包含的物质也将大大增多，密度大大提高了。形象地说，这时原子核是“沉浸于”电子中，常称之为“简并态”。[72]  大多数的恒星内核通过氢核聚变进行燃烧，将质量转变为能量，并产生光和热量，当恒星内部氢燃料完成消耗完后就开始进行氦融合反应，并形成更重的碳和氧，这一过程对于类似太阳这样的恒星而言，就显得较为短暂，并形成碳氧组成的白矮星，如果其质量大于1.4倍太阳质量，就会发生Ia型超新星爆发。[73] 类星体,20世纪60年代以来，天文学家还找到一种在银河系以外像恒星一样表现为一个光点的天体，但实际上它的光度和质量又和星系一样，我们叫它类星体，现在已发现了数千个这种天体。[74] 超新星，是恒星演化过程中的一个阶段。超新星爆发是某些恒星在演化接近末期时经历的一种剧烈爆炸。一般认为质量小于9倍太阳质量左右的恒星，在经历引力坍缩的过程后是无法形成超新星的。[75]  在大质量恒星演化到晚期，内部不能产生新的能量，巨大的引力将整个星体迅速向中心坍缩，将中心物质都压成中子状态，形成中子星，而外层下坍的物质遇到这坚硬的“中子核”反弹引起爆炸。这就成为超新星爆发，质量更大时，中心更可形成黑洞。[76]  在超新星爆发的过程中所释放的能量，需要我们的太阳燃烧900亿年才能与之相当。[77]  超新星研究有着关乎人类自身命运的深层意义。如果一颗超新星爆发的位置非常接近地球，目前国际天文学界普遍认为此距离在100光年以内，它就能够对地球的生物圈产生明显的影响，这样的超新星被称为近地超新星。有研究认为，在地球历史上的奥陶纪大灭绝，就是一颗近地超新星引起的，这次灭绝导致当时地球近60%的海洋生物消失

克莱因瓶是一个不可定向的二维紧流形，而球面或轮胎面是可 克莱因瓶 克莱因瓶 定向的二维紧流形。如果观察克莱因瓶，有一点似乎令人困惑－－ “67P/楚留莫夫-格拉希门克”彗星 [67] 在太 CHAPTER X. TOBACCO OFFERINGS. 1808. It was a beautiful moonlight evening in August. A shadowy haze lingered over the river, which glistened and sparkled in the moonlight. The Chief and several members of his family were seated on the beach in front of the Wigwam listening to the Honorable Joseph Papineau, who, with his son, Louis Joseph, had come up in a canoe to see the falls. The former had recently purchased from Bishop Laval the unsettled seigniory of Petit Nation, and had erected an unpretentious cottage, which he occupied during the summer months. HON. LOUIS JOSEPH PAPINEAU AND MADAME PAPINEAU. From Morgan's "Types of Canadian Women" (copyright, 1903), by permission. HON. LOUIS JOSEPH PAPINEAU AND MADAME PAPINEAU. From Morgan's "Types of Canadian Women" (copyright, 1903), by permission. "It was a lovely vision," said Mr. Papineau, who had just performed the feat of canoeing to the foot of the Chaudiere Falls for the first time. "On our return we climbed the rugged cliff on the south side, and never shall I forget the panorama that spread out before us. The sun, sinking slowly behind the Laurentian hills, had clothed himself with a robe of splendor. The long reflections lay soft on the waters of the river below. The clouds of ascending mist from the Chaudiere took a thousand shades of color as the western sky faded slowly from crimson into gold and from gold to green and gray, and finally displayed dark shapes, out of which imagination might well have formed a thousand monsters.* * Louis Joseph, afterwards known as the Demosthenes of Canada, and who almost succeeded in making Canada a Republic, with himself as President, was evidently much impressed with the scene, which he described as follows: "Le soleil etait pret decendre sous l'horison, la mureille tout limpide etait d'une transparence vivre, tout penetree de lumiere vaguement prismatiseé." "As we watched the gathering shadows my thoughts went back two hundred years, to the time when Champlain went on his first trip up the 'Riviere des Algoumequins,' as he called it. About two years before he took the trip he sent Nicholas de Vignan, a young Frenchman, up the river with some friendly Indians, and Nicholas had returned with the marvellous story that he had reached the North Sea. He said that the journey could be made in a few days. He also gave an account of having seen the wreck of an English ship. "Champlain was completely taken in, and lost no time in starting off to verify the discovery for which the world had been looking for some time. His fleet consisted of two canoes with two Indians and three Frenchmen, one of whom was De Vignan. It was in May, when the river was at its height. When they reached the Gatineau the Indians told him that their tribe were often compelled to conceal themselves amid the hills of the Upper Gatineau from their dreaded enemies, the Iroquois. When Champlain beheld the twin curtain falls yonder, 'like a slow dropping veil of the thinnest lawn,' he exclaimed, 'Le Rideau! Le Rideau!' The Indians told him that the waters formed an arcade under which they delighted to walk, and where they were only wet by the spray. As they rounded the lofty headland opposite he saw the cloud of mist rising from the falls, which the Indians called the 'Asticou,' which means 'Chaudiere' in French, or 'kettle' in English, for the water has worn out a deep basin into which it rushes with a whirling motion which boils up in the midst like a kettle. "You have probably been close enough to have seen it, Madame?" he said, addressing Mrs. Wright. "No," she replied, "I have always been too timid to venture so near to it in a canoe." "Champlain said," continued Mr. Papineau, "that he paddled as near as possible to the falls, when the Indians took the canoes and the Frenchmen and himself carried their arms and provisions. He described with great feeling the sharp and rugged rocks of the portages to pass the falls and rapids until at last, in the afternoon, they embarked upon the peaceful waters of a lake where, he said, there were very beautiful islands filled with vines and with walnut and other agreeable trees." "There are no walnuts on the islands of Lake Chaudiere," interrupted Bearie, "I am quite sure." "He probably saw a butternut tree," said young Louis Joseph, "and thought it produced walnuts." "Champlain's journey came to an abrupt close a few days afterwards," said Mr. Papineau, "when he reached Allumette Island, about seventy miles farther up the river. There was a large settlement of friendly Algonquins, called 'Les Sauvages de l'Isle,' and Champlain tried to obtain several canoes and guides to proceed farther. They, however, had their own commercial reasons for keeping the French from the upper country, and they warned him of the danger of meeting the terrible tribe of the Sorcerers. Champlain said that De Vignan had passed through all these dangers. The head Chief then said to the impostor: "'Is it true that you have said that you have been among the Sorcerers?' "After a long pause he said: 'Yes, I've been there.' "The Indians at once threw themselves upon him with fierce cries as if they would have torn him to pieces, and the Chief said: "'You are a bold liar. You know that every night you slept by my side with my children. How have you the impudence to tell your chief such lies?' "The upshot was that Champlain returned down the Ottawa, followed by an escort of fifty canoes. "When the party reached the Chaudiere the savages, he said, performed their mystic rites. After having carried their canoes to the foot of the Falls, they gathered in a certain spot where one of them, provided with a wooden dish, passed it round, and each one placed in the dish a piece of tobacco. "The collection finished, the dish was placed in the midst of the band and all danced around it, chanting after their fashion. Then one of the chiefs delivered a harangue, explaining that from olden times they had always made such an offering, and that by this means they are protected from their enemies and saved from misfortune, for so the devil persuades them. Then the same chief took the dish and proceeded to throw the tobacco into the Chaudiere, amid the loud shoutings of the band. 'They are so superstitious,' said Champlain, 'that they do not believe that they can make a safe journey if they have not performed this ceremony in this particular place.' "The Chief proceeded to throw the tobacco into the Chaudiere." "The Chief proceeded to throw the tobacco into the Chaudiere." "Ah, Monsieur," Mr. Papineau continued, "it stirred my soul as I stood on that rocky cliff and thought of how many canoes of heroic missionaries, Indian braves and cheery voyageurs have paddled these waters and torn their feet on the rocky shores, going, some of them to death and some to tortures worse than death. As we drifted down with the current in the moonlight the gentle breeze in the pines along the shore seemed to be whispering sad tales of other days." Mr. Papineau, who had spoken with such animation and fluency, relapsed into silence for several minutes, then, rousing himself, said, with even greater enthusiasm and vigor: "Providence has crowned our lives with great blessing since the heroic Daulac struck the death-blow to the power of the Iroquois in this country, and since the English undertook the responsibility of its government. Though I am proud of the fact that every bone and muscle, nerve and sinew within me is French, though I dearly love my Mother Country and my fellow countrymen, I have no hesitation in making the solemn assertion that our country has enjoyed a greater degree of prosperity under the new regime than it ever did under the old. But it must ever be remembered that much of the foundation of that prosperity was laid in the blood of the early French martyrs and in the heroic achievements of the early French settlers." It seemed incredible to the visitors that in a settlement of so recent date their host should have been able to show them a grist-mill, a saw-mill, a vegetable alkali factory, a tannery, a small foundry, a tailor shop, a bakery, a general store, and a hemp-mill, giving employment to over one hundred men. Fortunately for the pioneers of the Ottawa, they were not dependent upon the small revenue derived from the cultivation of the land, but had other resources which afforded them much greater remuneration. The British Navy, which hitherto had been dependent upon Russia for its cordage and lumber, had to look elsewhere for its supply of hemp and timber, owing to the ports of the Baltic having been closed to British ships. The price of hemp having risen from ￡25 to ￡118 per ton, they undertook the cultivation of it, and raised over three-fourths of the amount raised in Lower Canada at that time. The exportation of lumber and vegetable alkali, or potash, were also great sources of revenue. In the new clearances were tons of wood ashes from which the lye was extracted and boiled till it looked like molten iron, a barrel of which sold at that time for thirty dollars. Prosperity and success crowned every commercial enterprise upon which they ventured until fire swept every mill, factory and dwelling in the thriving little village out of existence, including thousands of dollars in cash in a small safe in the office, quantities of wheat, hemp, sawn lumber, laths and general merchandise. As there was no compensation in the way of insurance, the loss was much felt. Philemon Wright was not the man to be deterred from climbing the ladder of success, even though he had to mount it by the rungs of adverse circumstances. Though the loss sustained was great, almost overwhelming, he rose above it with a courage which yielded not to disappointment or failure. The cause of the fire long remained a mystery. That it was the work of an incendiary was beyond question. Various theories were advocated by the settlers, but suspicion rested upon Machecawa, who, it was alleged, had been seen by the bookkeeper at a late hour lingering about the mills, a suspicion which gained no credence with the Chief and his family.阳系的周围还包裹着一个庞大的“奥尔特云”。星云内分布着不计其数的冰块、雪团和碎石。其中的某些会受太阳引力影响飞入内太阳系，这就是彗 [76]  在超新星爆发的过程中所释放的能量，需要我们的太阳燃烧900亿年才能与之相当。[77]  超新星研究有着关乎人类自身命运的深层意义。如果一颗超新星爆发的位置非常接近地球，目前国际天文学界普遍认为此距离在100光年以内，它就能够对地球的生物圈产生明显的影响，这样的超新星被称为近地超新星。有研究认为，在地球历史上的奥陶纪大灭绝，就是一颗近地超新星引起的，这次灭绝导致当时地球近60%的海洋生物消失。[78]

克莱因瓶是一个不可定向的二维紧流形，而球面或轮胎面是可 克莱因瓶 克莱因瓶 定向的二维紧流形。如果观察克莱因瓶，有一点似乎令人困惑－－克莱因瓶的瓶颈和瓶身是相交的，换句话近代科学兴起的先驱者、是捍卫科学真理并为此献身的殉道士。有另一种说法认为，近代以来关于罗马梵蒂冈的地心说和哥白尼的日心说的斗争是被严重夸大的。布鲁诺1600年遭受火刑的原因，并非因知行星围绕太阳作圆周运动。然而，人们是否能接受哥白尼提出的新的宇宙模式呢？全世界的人——尤其是权力极大的天主教会是否相信太阳是宇宙中心这一说法呢？由于害怕教会的惩罚，哥白尼在世时不敢公开他的发现。1543年，这一发现才公诸天下。即使在那个时候，哥白尼的发现还不断受到教会高无上的真理，凡是违背圣经的学说，Eighteen months passed. The Chief was in Quebec with Hannah and Abbie awaiting the arrival of Rug, who had been sent by his father to the Mother Land to dispose of two cargoes of timber. It was an unusually cold evening in June. Snow had been falling all day. The neighboring hills were covered with large feathery crystals, which, however, soon melted as the sun appeared for a moment before sinking behind the gray walls of the Castle St. Louis. Just as the evening gun was fired, news had reached the union Hotel that a vessel had been sighted near the Island of Orleans. It was ascertained that it was the Dorris, in command of Captain French, and that Rug was on board. They were soon speeding down Mountain Street in a caleche to the docks, where they secured passage in a small row-boat which was going out to the vessel. The genial captain invited them to take tea with him, and said that Rug was below supervising and arranging with the Customs Officer about the baggage of his numerous protégés, and would be on deck shortly. Hannah burst into a paroxysm of tears when she caught sight of her long-lost lover, who had been compelled to leave only a few weeks after their marriage. He looked twenty years older, and appeared careworn, haggard and ill. As they were seated round the table he gave an account of his travels. "When I received your letter," he said, addressing his father, "I chartered two vessels and persuaded Archie and Jonathan Campbell to go with me for a pleasure trip. We were nearly three months tossing about at the mercy of wind and wave when a hurricane swept the deck of the vessel, carrying with it the main-mast and sails. Water began to pour in at an alarming rate, and after a desperate struggle at the pumps the captain ordered all hands on deck. We felt that we had to prepare for the worst. The sailors had abandoned the pumps from exhaustion, and Jonathan and I took their places and worked until we, too, were exhausted, and as others took our places we retired to the stern, where we found Archie in a sheltered nook, seated upon a coil of rope, playing his violin, apparently oblivious of our perilous condition. "For two days the work at the pumps was a matter of life and death, and when at last the wind subsided we drifted about helplessly until a passing vessel saw our signals of distress and towed us from the Bay of Biscay to Bristol, where the necessary repairs were made to enable us to proceed to Liverpool. We soon disposed of the timber at good profit, and Jonathan, Archie and I took the stage-coach for London, where we had the honor of being presented at Court to gay Prince Geordie, who is acting as Regent, owing to his father's mental derangement. I wish you could have seen the Carleton House," he said, turning to Hannah. "He built it at a cost of ￡250,000 sterling, and had to sell his stud of race-horses and discharge most of his servants to meet the demands of the creditors, for he had led such a wild, dissipated life that the King and Parliament refused for a long time to help him out of his difficulties. "We visited many places of interest in London and the old farm in Kent, which we found bordered on that of General Wolfe. Then we crossed to France, and after having with great difficulty secured passports, drove to Paris. "If we had arrived on the scene only a few months sooner we might have seen how Napoleon turned Louis XVIII. from the kingdom, or we might have seen the great battle of Waterloo; but Napoleon is now safe at St. Helena, where he was sent last October." "The story of Napoleon Bonaparte," said Captain French, "presents probably the most remarkable example in the world of the action of great intellect and resolute will, unrestrained by conscience, and shows both the possible success which may reward, for a time, the most unscrupulous selfishness and also, fortunately, its certain ultimate failure and overthrow." "Notwithstanding which, I have the greatest admiration for Napoleon," said Rug. "The Captain's sentiments are mine," said the Chief. "He was a man of no conscience, no heart, and one of the most uncompromising enemies of constitutional liberty that the world has ever seen. I am amazed that a born republican like you, Rug, could see anything to admire in despotism or tyranny." "Did you see anything of poor Josephine?" asked Abbie. "No," he said. "The Empress Queen Dowager died two years ago, but we saw her beautiful home, 'Malmaison.' "If one may judge from appearances, it will take many years for France to recover from the effects of the Reign of Terror. My object, however, in visiting France and England was that I might see something of their progressive developments in agriculture and commerce, so that we might adopt the newest and best methods in building up our own little colony. I have brought with me," he continued, "the latest novelties in the way of general merchandise; I have brought the newest inventions in agricultural and milling machinery; I have Herefordshire and Devon cattle, of most renowned ancestors, who have not ceased to protest against a sea voyage from the time they left Liverpool. "Nor is this all," he said; "I have something better still on board for the new settlement, namely, twenty-five English families, who are going to take up land in the township and pay for it in work." "And who nearly turned mutineers," added the captain, slapping him on the shoulder, "did they not, Wright?" "How was that?" asked the Chief. "When we boarded the vessel at Liverpool," replied Rug, "some were bright and cheerful, but most of them were in tears, which showed that they did not leave the Old Land without a struggle. We soon weighed anchor and were under sail with a fair wind, but it came round to the east and blew fresher, so that we were forced to come to anchor not far from the place we left. The ship, as you may see, was fitted up for the timber trade, and has only a small cabin or quarter-deck. On each side are ranged two tiers of berths for passengers providing their own bedding. Along the open space in the middle we placed two rows of large chests which were used sometimes as tables, sometimes as seats—all of which I shall show you presently. There was much noise and confusion before all found berths; crying children, swearing sailors, scolding women, who had not been able to secure the beds they wanted, produced a chorus of a very melancholy nature. The disagreeableness of it was heightened by the darkness of the night and the rolling and tossing of the ship. After breakfast, as usual, all began to be sick. I took the advice of the sailors and drank some salt water, which acted as an emetic, and I soon felt better. "Unfortunately, while we were still at anchor, boats came from the shore with friends of the sailors, who smuggled a lot of liquor on board, and before the captain discovered it the whole crew was drunk. We were wakened at an early hour next morning by the violent motion of the ship, for there was a perfect gale blowing from the north-west. The sea was roaring and foaming around us. The passengers were all sick. Things grew worse and worse. Consternation and alarm were in every face. Children were crying, women wringing their hands, and I could see by the angry looks of the men that they would like to have thrown me overboard. The ship had little ballast, and it mounted the waves like a feather. Sometimes a hard sea would break over her with a shock that would make every one stagger. After a sleepless night, in which I received many a bruise and uttered many a groan, the captain informed us that the squall had carried away our mainyard and rigging, and that we were on our way back to Bristol to refit. At one time, when the ship was on her side, several chests, though strongly lashed to the deck, broke from their moorings, and in their progress downwards carried destruction to everything on which they happened to fall. "What a sight the deck presented! Do you remember, Captain? Clothes, spoons, shoes, hats, bottles, dishes, were strewn about in endless confusion. The next day the captain returned with the mainyard dragging behind his boat, but owing to a strong head wind we could not prepare nor rig it till the following day, when all the men on board who could get round it assisted at the work, and we were soon speeding along at the rate of six miles an hour with a fine favorable breeze. "The next day we made one hundred miles in twelve hours. I cannot describe what took place after that, for I was too ill. It was well that I was ill, for the indignation of the men and the fury of the women were almost unbounded as they thought of having consented to leave their comfortable cottages to follow me to what I had represented was a new and better country. "As we neared the banks of Newfoundland a most extraordinary phenomenon was produced by the dashing of the salt water against the bow of the ship in the evening. The water seemed on fire and produced a very fine effect. The next day a mass of ice appeared about two hundred yards distant. It was almost half a mile in length, and was moving south-east. Soon after we found the channel between Cape Breton and Cape Ray, and got into the ice. The captain sent eight men to the bow with fenders. One piece knocked splinters off the bow and threw us all down. About five days later we reached the Island of Anticosti, but I was too ill to see it. We saw porpoises in shoals plunging about the ship, while the sailors tried to harpoon them beneath the bow. About two hundred and eighty miles below Quebec the pilot came on board. His number was painted in large characters on his sail as well as on his boat. He had a cask of fresh water and some maple sugar, which he sold at an extortionate price to the passengers. "Near Bic Island we saw whales spouting water at a great height, and a habitant came out in a boat with a large basket of eggs, which he disposed of at a shilling per dozen, and so we continued on until the domes and towers of Quebec came in sight and I began to realize the inexpressible joy of being at home once more."* * Diary of Rev. Robert Bell and letters of R. Wright. Rug was a young man of great executive ability, a young man whose word could be relied upon with absolute certainty, a young man who proved himself the very soul of honor in all his business transactions.都被斥为“异端邪说”，凡是反对神权统治的人，都被处以火刑。新兴的资产阶级为自己的生存和发展，掀起了一场反对封建制度和教会迷信思想的斗争，出现了人文主义的思潮。他们使用的战斗武器，就是未被神学染污的古希腊的哲学、科学和文艺。这就是震撼欧洲的文艺复兴运动。文艺复兴首先发生于意大利，很快就扩大到波兰及欧洲其他国家。与此同时，商业的活跃也促进了对外贸易的发展。在“黄金”这个符咒的驱使下，许多欧洲冒险者远航非洲、印度及整个远东地区。远洋航行需要丰富的天文和地理知识，从实际中积累起来的观测资料，使人们感到当时流行的“地静天动”的宇宙 CHAPTER IV. AN INDIAN SUITOR. 1803. Machecawa and his friend O'Jawescawa became frequent visitors at the Wigwam. They would come in the morning, uninvited, and sit silently all day long before the open fire and observe all that was going on. The spinning-wheel and hand-loom were objects of unceasing interest to them, and though it proved a great distraction to the children in their studies, and to the girls in the performance of their domestic duties, to have them there, they were always treated not only with respect but with consideration and kindness. One morning Machecawa stood gazing intently into the fire. His face wore an expression of perplexity. At length he turned to the White Chief, who was explaining a mathematical problem to one of his boys, and said: "Big Injun, he want to speak his thoughts from books. He want to know white man's Manitou." "May I teach him, father? Just for an hour every day?" said Chrissy, a tall, fair, thoughtful girl of seventeen, who was known throughout the settlement as the "Saint," for she had been led to take a serious view of life by a Quaker friend in the old school at Woburn. "It would be such a pleasure for me to lead him to a knowledge of the truth." The father readily granted the request, and it was arranged that he should receive instruction from Chrissy every morning while the younger boys were having their lessons. Never had teacher a more apt, humble, or willing pupil. Never had pupil a more considerate, patient, kind-hearted instructor. Over and over again did she repeat words and sentences until at last the Indian found, to his unspeakable joy, that he was beginning to acquire the words pretty freely. The morning hour with Machecawa proved of such interest that it was not an uncommon thing to see the White Chief and all the children listening intently to Chrissy and the Indian as they compared their respective creeds. One morning, after she had been giving an account of the creation and the deluge, she said, "Now, tell me what you think of these things. Do the Indians ever think of how the world was made? Did they ever hear of a flood?" Machecawa replied in broken English, the interpretation of which is as follows: The Indian believes that the great Manabozo is king of all other animal kings. The West Wind is his father, and his mother is grand-daughter of the Moon. Sometimes he is a wolf; sometimes a hare; sometimes he is a wicked spirit. Manabozo was hunting with his brother, a wolf, who fell through the ice in a lake and was eaten by snakes. Manabozo was very cross and changed himself into the stump of a tree and surprised the king of the serpents and killed him. The snakes were all Manitous, and they made the water flood the world. Manabozo climbed a tree which grew and grew as the flood came up and was saved from the wicked spirits. Manabozo looked over the waters and he saw a loon, and he cried to the loon for help to save the world. The loon went under the water to look for mud to build the world again, but he could not find the bottom. Then a muskrat tried, but he came up on his back nearly dead. Manabozo looked in his paws and found a little mud, and he took the mud and the dead body of the loon and with it created the world anew again. "And do you believe that?" said the White Chief. "Our tribe she believe like that," replied the Indian. "What is that thing tied round your neck, Machecawa?" said Bearie, the second son, a short, well knit, sturdy-looking youth of eighteen, whose every expression reflected a bright, happy, generous disposition. "She am my Manitou," replied the Indian. "What is a Manitou? Every Indian you meet with seems to differ on the subject." "Some tam she am wan ting, some tam she am anodder." "That is evading the question," said Chrissy. "What kind of a Manitou have you got inside of that little bag which is tied round your neck?" persisted Bearie. "Will you let me see it?" "No! No!! No!!!" he said excitedly. "My Manitou she am not be pleese." "Come, now, old man," he said. "Tell us all about it." "What is it?" "How did you get it?" "What is it for?" "Waal," he said, reluctantly, "When I am a boy, me, just become a man, my fadder, he say, 'Machecawa, tam you got a manitou.' My face he paint black, black. He say, heem, 'you no eat no teeng seex days.' By em by I am dream some teeng, me, dat some teeng she am my manitou. She help me kill beeg bear; she mak dem Iroquois dogs run like one wild moose. My fadder she am pleese; she make my manitou on my arm—see!" he said, rolling up his sleeve. On his shoulder was the rude outline of a fish, which had been tatooed with sharp bones and with the juice of berries rubbed in. "But what is in the little bag?" asked Bearie. "Will you let me see it?" After a good deal of reluctance he gave in at last, and two curious boys untied the precious parcel, while the others, equally curious, looked over his shoulders at a few old broken fish bones which were all the little bag contained. "Well, old man," said Bearie, slowly replacing the sacred relics, "we put our faith in something better than that. The white man trusts the Great Spirit in heaven to care for him and to take him to heaven when he dies." "Any bear in hebben?" asked the Indian. "No," said Bearie, "only good people." "Dat hebben she am no good for big Injun," said Machecawa, sadly. "De happy hunting ground she am full of moose, buffalo, bear, beaver. She am far, far away at de end of land, where de sun she sleep—two, tree moons away. One beeg dog she am cross, an' she bark at dead Injun, but he go on, an' on, an' on, an' den he am glad." It began to dawn upon the vigilant mother at length that it was not so much the wonders of civilization nor the desire to "speak his thoughts from books" that led Machecawa day after day to the Wigwam, as an ever-increasing interest in her fun-loving daughter, Abbie, who was a year younger than Chrissy, and who seemed unconscious of the fact that the eyes of the red chief were ever upon her.学说值得怀疑，这就要求人们进一步去探索宇宙的秘密，从而推进了天文学和地理学的发展。1492年，意大利著名的航海家哥伦布发现新大陆，麦哲伦和他的同伴绕地球一周，证明地球是圆形的，使人们开始真正认识地球。[4] 对他国的影响 在教会严密控制下的中世纪，也发生过轰轰烈烈的宗教革命。因为天主教的很多教义不符合圣经的教诲，而加入了太多教皇的个人意志以及各类神学家的自身成果，所以很多信徒开始质疑天主教的教义和组织，发起回归圣经的行动来。捷克的爱国主义者、布拉格大学校长扬·胡斯（1369～1415年）在君士坦丁堡的宗教会议上公开谴责德意志封建主与天主教会对捷克的压迫和剥削。他虽然被反动教会处以火刑，但他的革命活动在社会上引起了强烈的反应。捷克农民在胡斯党人的旗帜下举行起义，这次运动也波及波兰。1517年，在德国，马丁·路德（1483～1546年）反对教会贩卖赎罪符，与罗马教皇公开决裂。1521年，路德又在沃尔姆国会上揭露罗马教廷的罪恶，并提出建立基督教新教的主张。新教的教义得到许多国家的支持，波兰也深受影响。