1.5: Medical Language Within the Context of Anatomy and Physiology
- Connect medical language learning to the context of anatomy and physiology
- Introduce the basic architecture and levels of organization of the human body
- Evaluate the anatomical position, regional terms, directional terms, body planes, and body quadrants for anatomical positioning
- Describe body cavities and the functions of associated membranes
As you memorize the language components of medical terminology it is important to support that learning within the context of anatomy and physiology. Proceeding through the body system chapters you will learn word parts, whole medical terms, and common abbreviations. It is important to put into context where in the body the medical term is referencing, and then consider how it works within the body.
Anatomy focuses on structure and physiology focuses on function. Much of the study of physiology centers on the body’s tendency toward homeostasis .
Consider the structures of the body in terms of fundamental levels of organization that increase in complexity: subatomic particles, atoms, molecules, organelles, cells, tissues, organs, organ systems, organisms, and biosphere ( Figure 5.1 ).
The Levels of Organization
All matter in the universe is composed of one or more unique pure substances called elements , familiar examples are hydrogen, oxygen, carbon, nitrogen, calcium, and iron.
-
The smallest unit of any of these pure substances (elements) is an
atom
.
- Atoms are made up of subatomic particles such as the proton, electron, and neutron.
-
Two or more atoms combine to form a
molecule
, such as the water molecules, proteins, and sugars found in living things.
- Molecules are the chemical building blocks of all body structures.
-
A
cell
is
the smallest independently functioning unit of a living organism.
- Even bacteria, which are extremely small, independently-living organisms, have a cellular structure. Each bacterium is a single cell. All living structures of human anatomy contain cells, and almost all functions of human physiology are performed in cells or are initiated by cells
- A human cell typically consists of flexible membranes that enclose cytoplasm, a water-based cellular fluid, together with a variety of tiny functioning units called organelles .
- A tissue is a group of many similar cells (though sometimes composed of a few related types) that work together to perform a specific function.
- An organ is an anatomically distinct structure of the body composed of two or more tissue types. Each organ performs one or more specific physiological functions.
An organ system is a group of organs that work together to perform major functions or meet the physiological needs of the body .
Did you know?
- Organs are very collaborative and work with multiple body systems.
- F or example, the heart (cardiovascular system) and lungs (respiratory system) work together to deliver oxygen throughout the body and remove carbon dioxide from the body.
Consider the breakdown into eleven distinct organ systems in the human body ( Figure 5.2 and Figure 5.3 ). Assigning organs to organ systems can be imprecise since organs that “belong” to one system can also have functions integral to another system. In fact, most organs contribute to more than one system.
The organism level is the highest level of organization. An organism is a living being that has a cellular structure and that can independently perform all physiologic functions necessary for life. In multicellular organisms, including humans, all cells, tissues, organs, and organ systems of the body work together to maintain the life and health of the organism.
Watch this video:
Media 5.1. Introduction to Anatomy & Physiology: Crash Course A&P #1 [Online video]. Copyright 2015 by CrashCourse.
Terminology is used for the purpose of precision and reduction of medical error.
Anatomical Position
Anatomists and health care providers use terminology for the purpose of precision and to reduce medical errors. For example, is a scar “above the wrist” located on the forearm two or three inches away from the hand? Or is it at the base of the hand? Is it on the palm-side or back-side? By using precise anatomical terminology, we eliminate ambiguity. Anatomical terms derive from ancient Greek and Latin words.
To further increase precision, anatomists standardize the way in which they view the body. Just as maps are normally oriented with north at the top, the standard body “map,” or anatomical position , is that of the body standing upright, with the feet at shoulder width and parallel, toes forward. The upper limbs are held out to each side, and the palms of the hands face forward as illustrated.
Using this standard position reduces confusion. It does not matter how the body being described is oriented, the terms are used as if it is in anatomical position. For example, a scar in the “anterior (front) carpal (wrist) region” would be present on the palm side of the wrist. The term “anterior” would be used even if the hand were palm down on a table.
A body that is lying down is described as either prone or supine . These terms are sometimes used in describing the position of the body during specific physical examinations or surgical procedures.
Regional Terms
The human body’s numerous regions have specific terms to help increase precision. Notice that the term “brachium” or “arm” is reserved for the “upper arm” and “antebrachium” or “forearm” is used rather than “lower arm.” Similarly, “femur” or “thigh” is correct, and “leg” or “crus” is reserved for the portion of the lower limb between the knee and the ankle. You will be able to describe the body’s regions using the terms from the anatomical position.
Directional Terms
Directional terms are essential for describing the relative locations of different body structures. For instance, an anatomist might describe one band of tissue as “inferior to” another or a physician might describe a tumor as “superficial to” a deeper body structure. Commit these terms to memory to avoid confusion when you are studying or describing the locations of particular body parts.
- Anterior (or ventral ) describes the front or direction toward the front of the body. The toes are anterior to the foot.
- Posterior (or dorsal ) describes the back or direction toward the back of the body. The popliteus is posterior to the patella.
- Superior (or cranial ) describes a position above or higher than another part of the body proper. The orbits are superior to the oris.
- Inferior (or caudal ) describes a position below or lower than another part of the body proper; near or toward the tail (in humans, the coccyx, or lowest part of the spinal column). The pelvis is inferior to the abdomen.
- Lateral describes the side or direction toward the side of the body. The thumb (pollex) is lateral to the digits.
- Medial describes the middle or direction toward the middle of the body. The hallux is the medial toe.
- Proximal describes a position in a limb that is nearer to the point of attachment or the trunk of the body. The brachium is proximal to the antebrachium.
- Distal describes a position in a limb that is farther from the point of attachment or the trunk of the body. The crus is distal to the femur.
- Superficial describes a position closer to the surface of the body. The skin is superficial to the bones.
- Deep describes a position farther from the surface of the body. The brain is deep to the skull.
Practice these directional terms.
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- Find a partner and take turns choosing two body parts on your or your partner’s body.
- Using directional terms, describe the location of those body parts relative to one another.
Body Planes
A section is a two-dimensional surface of a three-dimensional structure that has been cut. Modern medical imaging devices enable clinicians to obtain “virtual sections” of living bodies. We call these scans. Body sections and scans can be correctly interpreted, however, only if the viewer understands the plane along which the section was made. A plane is an imaginary two-dimensional surface that passes through the body. There are three planes commonly referred to in anatomy and medicine:
- The sagittal plane is the plane that divides the body or an organ vertically into right and left sides. If this vertical plane runs directly down the middle of the body, it is called the midsagittal or median plane. If it divides the body into unequal right and left sides, it is called a parasagittal plane or less commonly a longitudinal section.
- The frontal plane is the plane that divides the body or an organ into an anterior (front) portion and a posterior (rear) portion. The frontal plane is often referred to as a coronal plane. (“Corona” is Latin for “crown.”)
- The transverse plane is the plane that divides the body or organ horizontally into upper and lower portions. Transverse planes produce images referred to as cross sections.
Can you locate the planes?
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Body Cavities and Serous Membranes
The body maintains its internal organization by means of membranes, sheaths, and other structures that separate compartments. The dorsal (posterior) cavity and the ventral (anterior) cavity are the largest body compartments ( Figure 5.6 ). These cavities contain and protect delicate internal organs, and the ventral cavity allows for significant changes in the size and shape of the organs as they perform their functions. The lungs, heart, stomach, and intestines, for example, can expand and contract without distorting other tissues or disrupting the activity of nearby organs.
Subdivisions of the Posterior (Dorsal) and Anterior (Ventral) Cavities
The posterior (dorsal) and anterior (ventral) cavities are each subdivided into smaller cavities:
The posterior (dorsal) cavity has two main subdivisions:
-
In the posterior (dorsal) cavity, the
cranial cavity
houses the brain
- Protected by the bones of the skulls and cerebrospinal fluid
-
The
spinal cavity
(or vertebral cavity) encloses the spinal cord.
- Protected by the vertebral column and cerebrospinal fluid
The anterior (ventral) cavity has two main subdivisions:
-
The
thoracic cavity
is the more superior subdivision of the anterior cavity, and it is enclosed by the rib cage.
- The thoracic cavity contains the lungs and the heart, which is located in the mediastinum.
- The diaphragm forms the floor of the thoracic cavity and separates it from the more inferior abdominopelvic cavity.
-
The
abdominopelvic cavity
is the largest cavity in the body.
- No membrane physically divides the abdominopelvic cavity.
- The abdominal cavity houses the digestive organs, the pelvic cavity, and the reproductive organs.
Practice locating cavities.
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Abdominal Regions and Quadrants
To promote clear communication, for instance about the location of a patient’s abdominal pain or a suspicious mass, health care providers typically divide up the cavity into either nine regions or four quadrants.
Practice locating the quadrants.
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Tissue Membranes
A tissue membrane is a thin layer or sheet of cells that covers the outside of the body (for example, skin), the organs (for example, pericardium), internal passageways that lead to the exterior of the body (for example, abdominal mesenteries), and the lining of the movable joint cavities. There are two basic types of tissue membranes: connective tissue and epithelial membranes ( Figure 5.7 ).
Synovial membranes line cavities which hold synovial fluid.
Synovial fluid lubricates the joints for movement.
Connective Tissue Membranes
-
The
connective tissue membrane
is formed solely from connective tissue.
- These membranes encapsulate organs, such as the kidneys, and line our movable joints.
-
A
synovial membrane
is a type of connective tissue membrane that lines the cavity of a freely movable joint.
- For example, synovial membranes surround the joints of the shoulder, elbow, and knee.
Epithelial Membranes
-
The
epithelial membrane
is composed of epithelium attached to a layer of connective tissue.
- For example, your skin.
-
The
mucous membrane
is also a composite of connective and epithelial tissues.
- Sometimes called mucosae, these epithelial membranes line the body cavities and hollow passageways that open to the external environment, and include the digestive, respiratory, excretory, and reproductive tracts.
- Mucus, produced by the epithelial exocrine glands, covers the epithelial layer.
- The underlying connective tissue, called the lamina propria (literally “own layer”), help support the fragile epithelial layer.
-
The skin is an epithelial membrane also called the
cutaneous membrane
.
- It is a stratified squamous epithelial membrane resting on top of connective tissue. The apical surface of this membrane is exposed to the external environment and is covered with dead, keratinized cells that help protect the body from desiccation and pathogens.
Membranes of the Anterior (Ventral) Body Cavity
-
A
serous membrane
(also referred to as serosa) is an epithelial membrane composed of mesodermally derived epithelium called the mesothelium that is supported by connective tissue. These membranes line the
coelomic
cavities of the body and they cover the organs located within those cavities. They are essentially membranous bags, with mesothelium lining the inside and connective tissue on the outside.
- Parietal layers: line the walls of the body cavity.
- Visceral layer: covers the organs (the viscera).
- Between the parietal and visceral layers is a very thin, fluid-filled serous space.
There are three serous cavities and their associated membranes. Serous membranes provide additional protection to the viscera they enclose by reducing friction that could lead to inflammation of the organs.
- Pleura : surrounds the lungs in the pleural cavity and reduces friction between the lungs and the body wall.
- Pericardium: surrounds the heart in the pericardial cavity and reduces friction between the heart and the wall of the pericardium.
- Peritoneum: surrounds several organs in the abdominopelvic cavity. The peritoneal cavity reduces friction between the abdominal and pelvic organs and the body wall.
Test Yourself
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References
[CrashCourse]. (2015, January 6). Introduction to anatomy & physiology: Crash course A&P #1 [Video]. YouTube. https://youtu.be/uBGl2BujkPQ
Image Descriptions
Figure 5.1 image description: This illustration shows biological organization as a pyramid. The chemical level is at the apex of the pyramid where atoms bond to form molecules with three dimensional structures. An example is shown with two white hydrogen atoms bonding to a red oxygen atom to create water. The next level down on the pyramid is the cellular level, as illustrated with a long, tapered, smooth muscle cell. At this level, a variety of molecules combine to form the interior fluid and organelles of a body cell. The next level down is the tissue level. A community of similar cells forms body tissue. The example given here is a section of smooth muscle tissue, which contains many smooth muscle cells closely bound side by side. The next level down is the organ level, as illustrated with the bladder and urethra. The bladder contains smooth muscle while the urethra contains skeletal muscle. These are both examples of muscle tissues. The next level down is the organ system level, as illustrated by the entire urinary system containing the kidney, ureters, bladder and urethra. At this level, two or more organs work closely together to perform the functions of a body system. At the base of the pyramid is the organismal level illustrated with a woman drinking water. At this level, many organ systems work harmoniously together to perform the functions of an independent organism. [Return to Figure 5.1] .
Figure 5.2 image description: This illustration shows eight silhouettes of a human female, each showing the components of a different organ system. The integumentary system encloses internal body structures and is the site of many sensory receptors. The integumentary system includes the hair, skin, and nails. The skeletal system supports the body and, along with the muscular system, enables movement. The skeletal system includes cartilage, such as that at the tip of the nose, as well as the bones and joints. The muscular system enables movement, along with the skeletal system, but also helps to maintain body temperature. The muscular system includes skeletal muscles, as well as tendons that connect skeletal muscles to bones. The nervous system detects and processes sensory information and activates bodily responses. The nervous system includes the brain, spinal cord, and peripheral nerves, such as those located in the limbs. The endocrine system secretes hormones and regulates bodily processes. The endocrine system includes the pituitary gland in the brain, the thyroid gland in the throat, the pancreas in the abdomen, the adrenal glands on top of the kidneys, and the testes in the scrotum of males as well as the ovaries in the pelvic region of females. The cardiovascular system delivers oxygen and nutrients to the tissues as well as equalizes temperature in the body. The cardiovascular system includes the heart and blood vessels. [Return to Figure 5.2] .
Figure 5.3 image description: The lymphatic system returns fluid to the blood and defends against pathogens. The lymphatic system includes the thymus in the chest, the spleen in the abdomen, the lymphatic vessels that spread throughout the body, and the lymph nodes distributed along the lymphatic vessels. The respiratory system removes carbon dioxide from the body and delivers oxygen to the blood. The respiratory system includes the nasal passages, the trachea, and the lungs. The digestive system processes food for use by the body and removes wastes from undigested food. The digestive system includes the stomach, the liver, the gall bladder (connected to the liver), the large intestine, and the small intestine. The urinary system controls water balance in the body and removes and excretes waste from the blood. The urinary system includes the kidneys and the urinary bladder. The reproductive system of males and females produce sex hormones and gametes. The male reproductive system is specialized to deliver gametes to the female while the female reproductive system is specialized to support the embryo and fetus until birth and produce milk for the infant after birth. The male reproductive system includes the two testes within the scrotum as well as the epididymis which wraps around each testis. The female reproductive system includes the mammary glands within the breasts and the ovaries and uterus within the pelvic cavity. [Return to Figure 5.3]
Figure 5.4 image description: This illustration shows an anterior and posterior view of the human body. The cranial region encompasses the upper part of the head while the facial region encompasses the lower half of the head beginning below the ears. The eyes are referred to as the ocular region. The cheeks are referred to as the buccal region. The ears are referred to as the auricle or otic region. The nose is referred to as the nasal region. The chin is referred to as the mental region. The neck is referred to as the cervical region. The trunk of the body contains, from superior to inferior, the thoracic region encompassing the chest, the mammary region encompassing each breast, the abdominal region encompassing the stomach area, the coxal region encompassing the belt line, and the pubic region encompassing the area above the genitals. The umbilicus, or naval, is located at the center of the abdomen. The pelvis and legs contain, from superior to inferior, the inguinal or groin region between the legs and the genitals, the pubic region surrounding the genitals, the femoral region encompassing the thighs, the patellar region encompassing the knee, the crural region encompassing the lower leg, the tarsal region encompassing the ankle, the pedal region encompassing the foot and the digital/phalangeal region encompassing the toes. The great toe is referred to as the hallux. The regions of the upper limbs, from superior to inferior, are the axillary region encompassing the armpit, the brachial region encompassing the upper arm, the antecubital region encompassing the front of the elbow, the antebrachial region encompassing the forearm, the carpal region encompassing the wrist, the palmar region encompassing the palm, and the digital/phalangeal region encompassing the fingers. The thumb is referred to as the pollux. The posterior view contains, from superior to inferior, the cervical region encompassing the neck, the dorsal region encompassing the upper back and the lumbar region encompassing the lower back. The regions of the back of the arms, from superior to inferior, include the cervical region encompassing the neck, acromial region encompassing the shoulder, the brachial region encompassing the upper arm, the olecranal region encompassing the back of the elbow, the antebrachial region encompasses the back of the arm, and the manual region encompassing the palm of the hand. The posterior regions of the legs, from superior to inferior, include the gluteal region encompassing the buttocks, the femoral region encompassing the thigh, the popliteus region encompassing the back of the knee, the sural region encompassing the back of the lower leg, and the plantar region encompassing the sole of the foot. Some regions are combined into larger regions. These include the trunk, which is a combination of the thoracic, mammary, abdominal, naval, and coxal regions. The cephalic region is a combination of all of the head regions. The upper limb region is a combination of all of the arm regions. The lower limb region is a combination of all of the leg regions. [Return to Figure 5.4].
Figure 5.5 image description: This illustration shows two diagrams: one of a side view of a female and the other of an anterior view of a female. Each diagram shows directional terms using double-sided arrows. The cranial-distal arrow runs vertically behind the torso and lower abdomen. The cranial arrow is pointing toward the head while the caudal arrow is pointing toward the tail bone. The posterior/anterior arrow is running horizontally through the back and chest. The posterior or dorsal arrow is pointing toward the back while the anterior, or ventral arrow, is pointing toward the abdomen. On the anterior view, the proximal/distal arrow is on the right arm. The proximal arrow is pointing up toward the shoulder while the distal arrow is pointing down toward the hand. The lateral-medial arrow is a horizontal arrow on the abdomen. The medial arrow is pointing toward the navel while the lateral arrow is pointing away from the body to the right. Right refers to the right side of the woman’s body from her perspective while left refers to the left side of the woman’s body from her perspective. [Return to Figure 5.5].
Figure 5.6 image description: This illustration shows a lateral and anterior view of the body and highlights the body cavities with different colors. The cranial cavity is a large, bean-shaped cavity filling most of the upper skull where the brain is located. The vertebral cavity is a very narrow, thread-like cavity running from the cranial cavity down the entire length of the spinal cord. Together the cranial cavity and vertebral cavity can be referred to as the dorsal body cavity. The thoracic cavity consists of three cavities that fill the interior area of the chest. The two pleural cavities are situated on both sides of the body, anterior to the spine and lateral to the breastbone. The superior mediastinum is a wedge-shaped cavity located between the superior regions of the two thoracic cavities. The pericardial cavity within the mediastinum is located at the center of the chest below the superior mediastinum. The pericardial cavity roughly outlines the shape of the heart. The diaphragm divides the thoracic and the abdominal cavities. The abdominal cavity occupies the entire lower half of the trunk, anterior to the spine. Just under the abdominal cavity, anterior to the buttocks, is the pelvic cavity. The pelvic cavity is funnel shaped and is located inferior and anterior to the abdominal cavity. Together the abdominal and pelvic cavity can be referred to as the abdominopelvic cavity while the thoracic, abdominal, and pelvic cavities together can be referred to as the ventral body cavity. [Return to Figure 5.6].
Figure 5.7 image description: This illustrations shows the silhouette of a human female from an anterior view. Several organs are showing in her neck, thorax, abdomen left arm and right leg. Text boxes point out and describe the mucous membranes in several different organs. The topmost box points to the mouth and trachea. It states that mucous membranes line the digestive, respiratory, urinary and reproductive tracts. They are coated with the secretions of mucous glands. The second box points to the outside edge of the lungs as well as the large intestine and states that serous membranes line body cavities that are closed to the exterior of the body, including the peritoneal, pleural and pericardial cavities. The third box points to the skin of the hand. It states that cutaneous membrane, also known as the skin, covers the body surface. The fourth box points to the right knee. It states that synovial membranes line joint cavities and produce the fluid within the joint. [Return to Figure 5.7]
Figure 5.8 image description: This diagram shows the pericardium on the left next to an analogy of a hand punching a balloon on the right. The pericardium is a two-layered sac that surrounds the entire heart except where the blood vessels emerge on the heart’s superior side. The pericardium has two layers because it folds over itself in the shape of the letter U. The inner layer that borders the heart is the visceral pericardium while the outer layer is the parietal pericardium. The space between the two layers is called the pericardial cavity. The heart sits in the cavity much like a fist punching into a balloon. The balloon surrounds the lower part of the fist with a two-layered sac, with the top of the balloon, where it contacts the fist, being analogous to the visceral pericardium. The bottom of the balloon, where it is tied off, is analogous to the parietal pericardium. The air within the balloon is analogous to the pericardial cavity. [Return to Figure 5.8].
Unless otherwise indicated, this chapter contains material adapted from Anatomy and Physiology (on OpenStax ), by Betts, et al. and is used under a a CC BY 4.0 international license . Download and access this book for free at https://openstax.org/books/anatomy-and-physiology/pages/1-introduction .


7.6: Connections of Carbohydrate, Protein, and Lipid Metabolic Pathways
Skills to Develop
- Discuss the ways in which carbohydrate metabolic pathways, glycolysis, and the citric acid cycle interrelate with protein and lipid metabolic pathways
- Explain why metabolic pathways are not considered closed systems
You have learned about the catabolism of glucose, which provides energy to living cells. But living things consume more than glucose for food. How does a turkey sandwich end up as ATP in your cells? This happens because all of the catabolic pathways for carbohydrates, proteins, and lipids eventually connect into glycolysis and the citric acid cycle pathways (see Figure \(\PageIndex{2}\)). Metabolic pathways should be thought of as porous—that is, substances enter from other pathways, and intermediates leave for other pathways. These pathways are not closed systems. Many of the substrates, intermediates, and products in a particular pathway are reactants in other pathways.
Connections of Other Sugars to Glucose Metabolism
Glycogen, a polymer of glucose, is an energy storage molecule in animals. When there is adequate ATP present, excess glucose is shunted into glycogen for storage. Glycogen is made and stored in both liver and muscle. The glycogen will be hydrolyzed into glucose monomers (G-1-P) if blood sugar levels drop. The presence of glycogen as a source of glucose allows ATP to be produced for a longer period of time during exercise. Glycogen is broken down into G-1-P and converted into G-6-P in both muscle and liver cells, and this product enters the glycolytic pathway.
Sucrose is a disaccharide with a molecule of glucose and a molecule of fructose bonded together with a glycosidic linkage. Fructose is one of the three dietary monosaccharides, along with glucose and galactose (which is part of the milk sugar, the disaccharide lactose), which are absorbed directly into the bloodstream during digestion. The catabolism of both fructose and galactose produces the same number of ATP molecules as glucose.
Connections of Proteins to Glucose Metabolism
Proteins are hydrolyzed by a variety of enzymes in cells. Most of the time, the amino acids are recycled into the synthesis of new proteins. If there are excess amino acids, however, or if the body is in a state of starvation, some amino acids will be shunted into the pathways of glucose catabolism (Figure \(\PageIndex{1}\)). Each amino acid must have its amino group removed prior to entry into these pathways. The amino group is converted into ammonia. In mammals, the liver synthesizes urea from two ammonia molecules and a carbon dioxide molecule. Thus, urea is the principal waste product in mammals produced from the nitrogen originating in amino acids, and it leaves the body in urine.
Connections of Lipid and Glucose Metabolisms
The lipids that are connected to the glucose pathways are cholesterol and triglycerides. Cholesterol is a lipid that contributes to cell membrane flexibility and is a precursor of steroid hormones. The synthesis of cholesterol starts with acetyl groups and proceeds in only one direction. The process cannot be reversed.
Triglycerides are a form of long-term energy storage in animals. Triglycerides are made of glycerol and three fatty acids. Animals can make most of the fatty acids they need. Triglycerides can be both made and broken down through parts of the glucose catabolism pathways. Glycerol can be phosphorylated to glycerol-3-phosphate, which continues through glycolysis. Fatty acids are catabolized in a process called beta-oxidation that takes place in the matrix of the mitochondria and converts their fatty acid chains into two carbon units of acetyl groups. The acetyl groups are picked up by CoA to form acetyl CoA that proceeds into the citric acid cycle.
Evolution Connection: Pathways of Photosynthesis and Cellular Metabolism
The processes of photosynthesis and cellular metabolism consist of several very complex pathways. It is generally thought that the first cells arose in an aqueous environment—a “soup” of nutrients—probably on the surface of some porous clays. If these cells reproduced successfully and their numbers climbed steadily, it follows that the cells would begin to deplete the nutrients from the medium in which they lived as they shifted the nutrients into the components of their own bodies. This hypothetical situation would have resulted in natural selection favoring those organisms that could exist by using the nutrients that remained in their environment and by manipulating these nutrients into materials upon which they could survive. Selection would favor those organisms that could extract maximal value from the nutrients to which they had access.
An early form of photosynthesis developed that harnessed the sun’s energy using water as a source of hydrogen atoms, but this pathway did not produce free oxygen (anoxygenic photosynthesis). (Early photosynthesis did not produce free oxygen because it did not use water as the source of hydrogen ions; instead, it used materials like hydrogen sulfide and consequently produced sulfur). It is thought that glycolysis developed at this time and could take advantage of the simple sugars being produced, but these reactions were unable to fully extract the energy stored in the carbohydrates. The development of glycolysis probably predated the evolution of photosynthesis, as it was well suited to extract energy from materials spontaneously accumulating in the “primeval soup.” A later form of photosynthesis used water as a source of electrons and hydrogen, and generated free oxygen. Over time, the atmosphere became oxygenated, but not before the oxygen released oxidized metals in the ocean and created a “rust” layer in the sediment, permitting the dating of the rise of the first oxygenic photosynthesizers. Living things adapted to exploit this new atmosphere that allowed aerobic respiration as we know it to evolve. When the full process of oxygenic photosynthesis developed and the atmosphere became oxygenated, cells were finally able to use the oxygen expelled by photosynthesis to extract considerably more energy from the sugar molecules using the citric acid cycle and oxidative phosphorylation.
Summary
The breakdown and synthesis of carbohydrates, proteins, and lipids connect with the pathways of glucose catabolism. The simple sugars are galactose, fructose, glycogen, and pentose. These are catabolized during glycolysis. The amino acids from proteins connect with glucose catabolism through pyruvate, acetyl CoA, and components of the citric acid cycle. Cholesterol synthesis starts with acetyl groups, and the components of triglycerides come from glycerol-3-phosphate from glycolysis and acetyl groups produced in the mitochondria from pyruvate.
Contributors and Attributions
-
Connie Rye (East Mississippi Community College), Robert Wise (University of Wisconsin, Oshkosh), Vladimir Jurukovski (Suffolk County Community College), Jean DeSaix (University of North Carolina at Chapel Hill), Jung Choi (Georgia Institute of Technology), Yael Avissar (Rhode Island College) among other contributing authors. Original content by OpenStax (CC BY 4.0; Download for free at<EMAIL_ADDRESS>).


20: Phylogenies and the History of Life
By following pathways of similarities and changes—both visible and genetic—scientists seek to map the evolutionary past of how life developed from single-celled organisms to the tremendous collection of creatures that have germinated, crawled, floated, swam, flown, and walked on this planet.
-
- 20.1: Organizing Life on Earth
- In scientific terms, the evolutionary history and relationship of an organism or group of organisms is called phylogeny. Phylogeny describes the relationships of an organism, such as from which organisms it is thought to have evolved, to which species it is most closely related, and so forth. Phylogenetic relationships provide information on shared ancestry but not necessarily on how organisms are similar or different.
-
- 20.2: Determining Evolutionary Relationships
- Scientists must collect accurate information that allows them to make evolutionary connections among organisms. Similar to detective work, scientists must use evidence to uncover the facts. In the case of phylogeny, evolutionary investigations focus on two types of evidence: morphologic (form and function) and genetic.
-
- 20.3: Perspectives on the Phylogenetic Tree
- The concepts of phylogenetic modeling are constantly changing. It is one of the most dynamic fields of study in all of biology. Over the last several decades, new research has challenged scientists’ ideas about how organisms are related. New models of these relationships have been proposed for consideration by the scientific community.


2.1: Melting Point Analysis
Melting point (Mp) is a quick and easy analysis that may be used to qualitatively identify relatively pure samples (approximately <10% impurities). It is also possible to use this analysis to quantitatively determine purity. Melting point analysis, as the name suggests, characterizes the melting point, a stable physical property, of a sample in a straightforward manner, which can then be used to identify the sample.
Equipment
Although different designs of apparatus exist, they all have some sort of heating or heat transfer medium with a control, a thermometer, and often a backlight and magnifying lens to assist in observing melting ( Figure \(\PageIndex{1}\) ). Most models today utilize capillary tubes containing the sample submerged in a heated oil bath. The sample is viewed with a simple magnifying lens. Some new models have digital thermometers and controls and even allow for programming. Programming allows more precise control over the starting temperature, ending temperature and the rate of change of the temperature.
Sample Preparation
For melting point analysis, preparation is straight forward. The sample must be thoroughly dried and relatively pure ( <10% impurities). The dry sample should then be packed into a melting point analysis capillary tube, which is simply a glass capillary tube with only one open end. Only 1 to 3 mm of sample is needed for sufficient analysis. The sample needs to be packed down into the closed end of the tube. This may be done by gently tapping the tube or dropping it upright onto a hard surface ( Figure \(\PageIndex{2}\) ). Some apparatuses have a vibrator to assist in packing the sample. Finally the tube should be placed into the machine. Some models can accommodate multiple samples.
Recording Data
Performing analysis is different from machine to machine, but the overall process is the same ( Figure \(\PageIndex{3}\) ). If possible, choose a starting temperature, ending temperature, and rate of change of temperature. If the identity of the sample is known, base the starting and ending temperatures from the known melting point of the chemical, providing margins on both sides of the range. If using a model without programming, simply turn on the machine and monitor the rate of temperature change manually.
Figure \(\PageIndex{3}\) A video discussing sample preparation, recording data and melting point analysis in general. Made by Indiana University-Purdue University Indianapolis chemistry department.
Visually inspect the sample as it heats. Once melting begins, note the temperature. When the sample is completely melted, note the temperature again. That is the melting point range for the sample. Pure samples typically have a 1 - 2 °C melting point range, however, this may be broadened due to colligative properties.
Interpreting Data
There are two primary uses of melting point analysis data. The first is for qualitative identification of the sample, and the second is for quantitative purity characterization of the sample.
For identification, compare the experimental melting point range of the unknown to literature values. There are several vast databases of these values. Obtain a pure sample of the suspected chemical and mix a small amount of the unknown with it and conduct melting point analysis again. If a sharp melting point range is observed at similar temperatures to the literature values, then the unknown has likely been identified correctly. Conversely, if the melting point range is depressed or broadened, which would be due to colligative properties, then the unknown was not successfully identified.
To characterize purity, first the identity of the solvent (the main constituent of the sample) and the identity of the primary solute need to be known. This may be done using other forms of analysis, such as gas chromatography-mass spectroscopy coupled with a database. Because melting point depression is unique between chemicals, a mixed melting curve comparing molar fractions of the two constituents with melting point needs to either be obtained or prepared ( Figure \(\PageIndex{4}\) ). Simply prepare standards with known molar fraction ratios, then perform melting point analysis on each standard and plot the results. Compare the melting point range of the experimental sample to the curve to identify the approximate molar fractions of the constituents. This sort of purity characterization cannot be performed if there are more than two primary components to the sample.
Specificity and Accuracy
Melting point analysis is fairly specific and accurate given its simplicity. Because melting point is a unique physical characteristic of a substance, melting point analysis does have high specificity. Although, many substances have similar melting points, so having an idea of possible chemicals in mind can greatly narrow down the choices. The thermometers used are also accurate. However, melting point is dependent on pressure as well, so experimental results can vary from literature values, especially at extreme locations, i.e., places of high altitude. The biggest source of error stems from the visual detection of melting by the experimenter. Controlling the change rate and running multiple trials can lessen the degree of error introduced at this step.
Advantages of Melting Point Analysis
Melting point analysis is a quick, relatively easy, and inexpensive preliminary analysis if the sample is already mostly pure and has a suspected identity. Additionally, analysis requires small samples only.
Limitations of Melting Point Analysis
As with any analysis, there are certain drawbacks to melting point analysis. If the sample is not solid, melting point analysis cannot be done. Also, analysis is destructive of the sample. For qualitative identification analysis, there are now more specific and accurate analyses that exist, although they are typically much more expensive. Also, samples with more than one solute cannot be analyzed quantitatively for purity.


14.5: On Inversive Transformations
Recall that the inversive plane is the Euclidean plane with an added point at infinity, denoted by \(\infty\) . We assume that every line passes thru \(\infty\) . Recall that the term circline stands for circle or line .
An inversive transformation is a bijection from the inversive plane to itself that sends circlines to circlines. Inversive geometry studies the circline incidence structure of the inversive plane (it sees which points lie on which circlines and nothing else).
A map from the inversive plane to itself is an inversive transformation if and only if it can be presented as a composition of inversions and reflections.
Exercise 18.8.3 gives another description of inversive transformations by means of complex coordinates.
- Proof
-
Evidently, reflection is an inversive transformation — it maps lines to lines and circles to circles. According to Theorem 10.3.1 , any inversion is an inversive transformation as well. Therefore, the same holds for any composition of inversions and reflections.
To prove the converse, fix an inversive transformation \(\alpha\) .
Assume \(\alpha(\infty)=\infty\) . Recall that any circline passing thru \(\infty\) is a line. If follows that \(\alpha\) maps lines to lines; that is, \(\alpha\) is an affine transformation that also maps circles to circles.
Note that any motion or scaling (defined in Exercise 14.1.2b ) are affine transformations that map circles to circles. Composing \(\alpha\) with motions and scalings, we can obtain another affine transformation \(\alpha'\) that maps a given unit circle \(\Gamma\) to itself. By Exercise 14.2.3 , \(\alpha'\) fixes the center, say \(O\) , of the circle \(\Gamma\) .
Set \(P'=\alpha'(P)\) . It follows that if \(OP=1\) , then \(OP'=1\) . By Proposition 14.3.1 , \(OP=OP'\) for any point \(P\) . Finally, by Exercise 14.3.1 , we have that if \(\overrightarrow{XY} = \overrightarrow{OP}\) , then \(\overrightarrow{X'Y'}=\overrightarrow{O'P'}\) . It follows that \(XY=X'Y'\) for any points \(X\) and \(Y\) ; that is, \(\alpha'\) is a motion.
Summarizing the discussion above, \(\alpha\) is a composition of motions and scalings. Observe that any scaling is a composition of two inversions in concentric circles. Recall that any motion is a composition of reflections (see Exercise 5.4.1 ). Whence \(\alpha\) is a composition of inversions and reflections.
In the remaining case \(\alpha(\infty) \ne \infty\) , set \(P=\alpha(\infty)\) . Consider an inversion \(\beta\) in a circle with center at \(P\) and set \(\gamma=\beta \circ \alpha\) . Note that \(\beta(P)=\infty\) ; therefore, \(\gamma(\infty)=\infty\) . Since \(\alpha\) and \(\beta\) are inversive, so is \(\gamma\) . From above we get that \(\gamma\) is a composition of reflections and inversions. Since \(\beta\) is self-inverse, we get \(\alpha= \beta \circ \gamma\) ; therefore \(\alpha\) is a composition of reflections and inversions as well.
Show that inversive transformations preserve the angle between arcs up to sign.
 Let \(\alpha\) and \(\alpha'\) denote the angle between the tangent half-lines to \(AB_1C_1\) and \(AB_2C_2\) at \(A\) and the angle between the tangent half-lines to \(A'B_1'C_1'\) and \(A'B_2'C_2'\) at \(A'\) respectively. Then
\(\alpha'=\pm \alpha.\)
- Hint
-
Apply Theorem 10.6.1 and Theorem \(\PageIndex{1}\)
Show that any reflection can be presented as a composition of three inversions.
- Hint
-
Fix a line \(\ell\). Choose a circle \(\Gamma\) with its center not on \(\ell\). Let \(\Omega\) be the inverse of \(\ell\) in \(\Gamma\); not that \(\Omega\) is a circle.
Let \(\iota_{\Gamma}\) and \(\iota_{\Omega}\) dentoe the inversions in \(\Gamma\) and \(\Omega\). Apply Corollary 10.6.1 to show that the composition \(\iota_{\Gamma} \circ \iota_{\Omega} \circ \iota_{\Gamma}\) is the reflection across \(\ell\).
The exercise above implies a stronger version of Theorem \(\PageIndex{1}\); namely any inversive transformation is a composition of inversions — no reflections needed.


10.6: Mean Ionic Activity Coefficients from Osmotic Coefficients
Recall that \(\g_{\pm}\) is the mean ionic activity coefficient of a strong electrolyte, or the stoichiometric activity coefficient of an electrolyte that does not dissociate completely.
The general procedure described in this section for evaluating \(\g_{\pm}\) requires knowledge of the osmotic coefficient \(\phi_m\) as a function of molality. \(\phi_m\) is commonly evaluated by the isopiestic method (Sec. 9.6.4) or from measurements of freezing-point depression (Sec. 12.2).
The osmotic coefficient of a binary solution of an electrolyte is defined by \begin{gather} \s{ \phi_m \defn \frac{\mu\A^*-\mu\A}{RTM\A\nu m\B} } \tag{10.6.1} \cond{(binary electrolyte solution)} \end{gather} That is, for an electrolyte the sum \(\sum_{i\neq \tx{A}}m_i\) appearing in the definition of \(\phi_m\) for a nonelectrolyte solution (Eq. 9.6.11) is replaced by \(\nu m\B\), the sum of the ion molalities assuming complete dissociation. It will now be shown that \(\phi_m\) defined this way can be used to evaluate \(\g_{\pm}\).
The derivation is like that described in Sec. 9.6.3 for a binary solution of a nonelectrolyte. Solving Eq. 10.6.1 for \(\mu\A\) and taking the differential of \(\mu\A\) at constant \(T\) and \(p\), we obtain \begin{equation} \dif\mu\A = -RTM\A\nu(\phi_m\dif m\B + m\B\dif\phi_m) \tag{10.6.2} \end{equation} From Eq. 10.3.9, we obtain \begin{equation} \dif\mu\B = RT\nu\left(\dif\ln\g_{\pm} + \frac{\dif m\B}{m\B}\right) \tag{10.6.3} \end{equation} Substitution of these expressions in the Gibbs–Duhem equation \(n\A \dif\mu\A + n\B \dif\mu\B = 0\), together with the substitution \(n\A M\A = n\B/m\B\), yields \begin{equation} \dif\ln\g_{\pm} = \dif\phi_m + \frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.4} \end{equation} Then integration from \(m\B = 0\) to any desired molality \(m'\B\) gives the result \begin{equation} \ln\g_{\pm}(m'\B) = \phi_m (m'\B) - 1 + \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.5} \end{equation} The right side of this equation is the same expression as derived for \(\ln\g\mbB\) for a nonelectrolyte (Eq. 9.6.20).
The integrand of the integral on the right side of Eq. 10.6.5 approaches \(-\infty\) as \(m\B\) approaches zero, making it difficult to evaluate the integral by numerical integration starting at \(m\B = 0\). (This difficulty does not exist when the solute is a nonelectrolyte.) Instead, we can split the integral into two parts \begin{equation} \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.6} \end{equation} where the integration limit \(m''\B\) is a low molality at which the value of \(\phi_m\) is available and at which \(\g_{\pm}\) can either be measured or estimated from the Debye–Hückel equation.
We next rewrite Eq. 10.6.5 with \(m\B'\) replaced with \(m\B''\): \begin{equation} \ln\g_{\pm}(m''\B) = \phi_m (m''\B) - 1 + \int_{0}^{m''\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.7} \end{equation} By eliminating the integral with an upper limit of \(m\B''\) from Eqs. 10.6.6 and 10.6.7, we obtain \begin{equation} \int_{0}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B = \ln\g_{\pm}(m''\B) - \phi_m (m''\B) + 1 + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.8} \end{equation} Equation 10.6.5 becomes \begin{equation} \ln\g_{\pm}(m'\B) = \phi_m(m'\B) - \phi_m(m''\B) + \ln\g_{\pm}(m''\B) + \int_{m''\B}^{m'\B}\frac{\phi_m - 1}{m\B}\dif m\B \tag{10.6.9} \end{equation} The integral on the right side of this equation can easily be evaluated by numerical integration.


10.1: Functions of a Random Variable
Introduction
Frequently, we observe a value of some random variable, but are really interested in a value derived from this by a function rule. If \(X\) is a random variable and \(g\) is a reasonable function (technically, a Borel function ), then \(Z = g(X)\) is a new random variable which has the value \(g(t)\) for any \(\omega\) such that \(X(\omega) = t\). Thus \(Z(\omega) = g(X(\omega))\).
The problem; an approach
We consider, first, functions of a single random variable. A wide variety of functions are utilized in practice.
Example 10.1 .1: A quality control problem
In a quality control check on a production line for ball bearings it may be easier to weigh the balls than measure the diameters. If we can assume true spherical shape and \(w\) is the weight, then diameter is \(kw^{1/3}\), where \(k\) is a factor depending upon the formula for the volume of a sphere, the units of measurement, and the density of the steel. Thus, if \(X\) is the weight of the sampled ball, the desired random variable is \(D = kX^{1/3}\).
Example 10.1.2: Price breaks
The cultural committee of a student organization has arranged a special deal for tickets to a concert. The agreement is that the organization will purchase ten tickets at $20 each (regardless of the number of individual buyers). Additional tickets are available according to the following schedule:
- 11-20, $18 each
- 21-30, $16 each
- 31-50, $15 each
- 51-100, $13 each
If the number of purchasers is a random variable \(X\), the total cost (in dollars) is a random quantity \(Z = g(X)\) described by
\(g(X) = 200 + 18 I_{M1} (X) (X - 10) + (16 - 18) I_{M2} (X) (X - 20)\)
\(+ (15 - 16) I_{M3} (X) (X - 30) + (13 - 15) I_{M4} (X) (X - 50)\)
where \(M1 = [10, \infty)\), \(M2 = [20, \infty)\), \(M3 = [30, \infty)\), \(M4 = [50, \infty)\)
The function rule is more complicated than in Example 10.1.1, but the essential problem is the same.
The problem
If \(X\) is a random variable, then \(Z = g(X)\) is a new random variable. Suppose we have the distribution for \(X\). How can we determine \(P(Z \in M)\), the probability \(Z\) takes a value in the set \(M\)?
An approach to a solution
We consider two equivalent approaches
To find \(P(X \in M)\).
-
Mapping approach
. Simply find the amount of probability mass mapped into the set \(M\) by the random variable \(X\).
- In the absolutely continuous case, calculate \(\int_{M} f_X\).
- In the discrete case, identify those values \(t_i\) of \(X\) which are in the set \(M\) and add the associated probabilities.
- Discrete alternative . Consider each value \(t_i\) of \(X\). Select those which meet the defining conditions for \(M\) and add the associated probabilities. This is the approach we use in the MATLAB calculations. Note that it is not necessary to describe geometrically the set \(M\); merely use the defining conditions.
To find \(P(g(X) \in M)\).
-
Mapping approach
. Determine the set \(N\) of all those
t
which are mapped into \(M\) by the function \(g\). Now if \(X(\omega) \in N\), then \(g(X(\omega)) \in M\), and if \(g(X(\omega)) \in M\), then \(X(\omega) \in N\). Hence
\(\{\omega: g(X(\omega)) \in M\} = \{\omega: X(\omega) \in N\}\)
Since these are the same event, they must have the same probability. Once \(N\) is identified, determine \(P(X \in N)\) in the usual manner (see part a, above).
- Discrete alternative . For each possible value \(t_i\) of \(X\), determine whether \(g(t_i)\) meets the defining condition for \(M\). Select those \(t_i\) which do and add the associated probabilities.
— □
Remark . The set \(N\) in the mapping approach is called the inverse image \(N = g^{-1} (M)\)
Example 10.1.3: A discrete example
Suppose \(X\) has values -2, 0, 1, 3, 6, with respective probabilities 0.2, 0.1, 0.2, 0.3 0.2.
Consider \(Z = g(X) = (X + 1) (X - 4)\). Determine \(P(Z > 0)\).
Solution
First solution . The mapping approach
\(g(t) = (t + 1) (t - 4)\). \(N = \{t: g(t) > 0\}\) is the set of points to the left of –1 or to the right of 4. The \(X\)-values –2 and 6 lie in this set. Hence
\(P(g(X) > 0) = P(X = -2) + P(X = 6) = 0.2 + 0.2 = 0.4\)
Second solution . The discrete alternative
| X = | -2 | 0 | 1 | 3 | 6 |
| P X = | 0.2 | 0.1 | 0.2 | 0.3 | 0.2 |
| Z = | 6 | -4 | -6 | -4 | 14 |
| Z > 0 | 1 | 0 | 0 | 0 | 1 |
Picking out and adding the indicated probabilities, we have
\(P(Z > 0) = 0.2 + 0.2 = 0.4\)
In this case (and often for “hand calculations”) the mapping approach requires less calculation. However, for MATLAB calculations (as we show below), the discrete alternative is more readily implemented.
Example 10.1.4 . An absolutely continuous example
Suppose \(X\) ~ uniform [–3,7]. Then \(f_X(t) = 0.1\), \(-3 \le t \le 7\) (and zero elsewhere). Let
\(Z = g(X) = (X + 1) (X - 4)\)
Determine \(P(Z > 0)\).
Solution
First we determine \(N = \{t: g(t) > 0\}\). As in Example 10.1.3, \(g(t) = (t+ 1) (t - 4) > 0\) for \(t < -1\) or \(t > 4)\). Because of the uniform distribution, the integral of the density over any subinterval of \(\{X, Y\}\) is 0.1 times the length of that subinterval. Thus, the desired probability is
\(P(g(X) > 0) = 0.1 [(-1 - (-3)) + (7 - 4)] = 0.5\)
We consider, next, some important examples.
Example 10.1.5: The normal distribution and standardized normal distribution
To show that if \(X\) ~ \(N(\mu, \sigma^2)\) then
\(Z = g(X) = \dfrac{X - \mu}{\sigma} ~ N(0, 1)\)
VERIFICATION
We wish to show the denity function for \(Z\) is
\(\varphi (t) = \dfrac{1}{\sqrt{2\pi}} e^{-t^2/2}\)
Now
\(g(t) = \dfrac{t - \mu} {\sigma} \le v\) iff \(t \le \sigma v + \mu\)
Hence, for given \(M = (-\infty, v]\) the inverse image is \(N = (-\infty, \sigma v + \mu]\), so that
\(F_Z (v) = P(Z \le v) = P(Z \in M) = P(X \in N) = P(X \le \sigma v + \mu) = F_X (\sigma v + \mu)\)
Since the density is the derivative of the distribution function,
\(f_Z(v) = F_{Z}^{'} (v) = F_{X}^{'} (v) = F_{X}^{'} (\sigma v + \mu) \sigma = \sigma f_X (\sigma v + \mu)\)
Thus
\(f_Z (v) = \dfrac{\sigma}{\sigma \sqrt{2\pi}} \text{exp} [-\dfrac{1}{2} (\dfrac{\sigma v + \mu - \mu}{\sigma})^2] = \dfrac{1}{\sqrt{2\pi}} e^{-v^2/2} = \varphi(v)\)
We conclude that \(Z\) ~ \(N(0, 1)\).
Example \(\PageIndex{1}\)
Suppose \(X\) has distribution function \(F_X\). If it is absolutely continuous, the corresponding density is \(f_X\). Consider \(Z = aX + b\). Here \(g(t) = at + b\), an affine function (linear plus a constant). Determine the distribution function for \(Z\) (and the density in the absolutely continuous case).
Solution
\(F_Z (v) = P(Z \le v) = P(aX + b \le v)\)
There are two cases
-
\(a\) > 0:
\(F_Z (v) = P(X \le \dfrac{v - b}{a}) = F_X (\dfrac{v - b}{a})\)
-
\(a\) < 0
\(F_Z (v) = P(X \ge \dfrac{v - b}{a}) = P(X > \dfrac{v - b}{a}) + P(X = \dfrac{v - b}{a})\)
So that
\(F_Z (v) = 1 - F_X (\dfrac{v - b}{a}) + P(X = \dfrac{v - b}{a})\)
For the absolutely continuous case, \(P(X = \dfrac{v - b}{a}) = 0\), and by differentiation
- for \(a > 0\) \(f_Z (v) = \dfrac{1}{a} f_X (\dfrac{v - b}{a})\)
- for \(a < 0\) \(f_Z (v) = -\dfrac{1}{a} f_X (\dfrac{v - b}{a})\)
Since for \(a < 0\), \(-a = |a|\), the two cases may be combined into one formula.
\(f_Z (v) = \dfrac{1}{|a|} f_X (\dfrac{v-b}{a})\)
Example 10.1.7: Completion of normal and standardized normal relationship
Suppose \(Z\) ~ \(N(0, 1)\). show that \(X = \sigma Z + \mu \) (\(\sigma > 0\)) is \(N(\mu, \sigma^2)\).
VERIFICATION
Use of the result of Example 10.1.6 on affine functions shows that
\(f_{X} (t) = \dfrac{1}{\sigma} \varphi (\dfrac{t - \mu}{\sigma}) = \dfrac{1}{\sigma \sqrt{2\pi}} \text{exp} [-\dfrac{1}{2} (\dfrac{t - \mu}{\sigma})^2]\)
Example 10.1.8: Fractional power of a nonnegative random variable
Suppose \(X \ge 0\) and \(Z = g(X) = X^{1/a}\) for \(a > 1\). Since for \(t \ge 0\), \(t^{1/a}\) is increasing, we have \(0 \le t^{1/a} \le v\) iff \(0 \le t \le v^{a}\). Thus
\(F_Z (v) = P(Z \le v) = P(X \le v^{a}) = F_X (v^{a})\)
In the absolutely continuous case
\(f_Z (v) = F_{Z}^{'} (v) = f_X (v^{a}) a v^{a - 1}\)
Example 10.1.9: Fractional power of an exponentially distributed random variable
Suppose \(X\) ~ exponential (\(\lambda\)). Then \(Z = X^{1/a}\) ~ Weibull \((a, \lambda, 0)\).
According to the result of Example 10.1.8,
\(F_Z(t) = F_X (t^{a}) = 1- e^{-\lambda t^{a}}\)
which is the distribution function for \(Z\) ~ Weibull \((a, \lambda, 0)\).
Example 10.1.10: A simple approximation as a function of X
If \(X\) is a random variable, a simple function approximation may be constructed (see Distribution Approximations). We limit our discussion to the bounded case, in which the range of \(X\) is limited to a bounded interval \(I = [a, b]\). Suppose \(I\) is partitioned into \(n\) subintervals by points \(t_i\), \(1 \le i \le n - 1\), with \(a = t_0\) and \(b = t_n\). Let \(M_i = [t_{i - 1}, t_i)\) be the \(i\)th subinterval, \(1 \le i \le n- 1\) and \(M_n = [t_{n -1}, t_n]\). Let \(E_i = X^{-1} (M_i)\) be the set of points mapped into \(M_i\) by \(X\). Then the \(E_i\) form a partition of the basic space \(\Omega\). For the given subdivision, we form a simple random variable \(X_s\) as follows. In each subinterval, pick a point \(s_i, t_{i - 1} \le s_i < t_i\). The simple random variable
\(X_s = \sum_{i = 1}^{n} s_i I_{E_i}\)
approximates \(X\) to within the length of the largest subinterval \(M_i\). Now \(I_{E_i} = I_{M_i} (X)\), since \(I_{E_i} (\omega) = 1\) iff \(X(\omega) \in M_i\) iff \(I_{M_i} (X(\omega)) = 1\). We may thus write
\(X_s = \sum_{i = 1}^{n} s_i I_{M_i} (X)\), a function of \(X\)
Use of MATLAB on simple random variables
For simple random variables, we use the discrete alternative approach, since this may be implemented easily with MATLAB. Suppose the distribution for \(X\) is expressed in the row vectors \(X\) and \(PX\).
-
We perform
array operations
on vector \(X\) to obtain
\(G = [g(t_1) g(t_2) \cdot\cdot\cdot g(t_n)]\)
- We use relational and logical operations on \(G\) to obtain a matrix \(M\) which has ones for those \(t_i\) (values of \(X\)) such that \(g(t_i)\) satisfies the desired condition (and zeros elsewhere).
- The zero-one matrix \(M\) is used to select the the corresponding \(p_i = P(X = t_i)\) and sum them by the taking the dot product of \(M\) and \(PX\).
Example 10.1.11: Basic calculations for a function of a simple random variable
X = -5:10; % Values of X PX = ibinom(15,0.6,0:15); % Probabilities for X G = (X + 6).*(X - 1).*(X - 8); % Array operations on X matrix to get G = g(X) M = (G > - 100)&(G < 130); % Relational and logical operations on G PM = M*PX' % Sum of probabilities for selected values PM = 0.4800 disp([X;G;M;PX]') % Display of various matrices (as columns) -5.0000 78.0000 1.0000 0.0000 -4.0000 120.0000 1.0000 0.0000 -3.0000 132.0000 0 0.0003 -2.0000 120.0000 1.0000 0.0016 -1.0000 90.0000 1.0000 0.0074 0 48.0000 1.0000 0.0245 1.0000 0 1.0000 0.0612 2.0000 -48.0000 1.0000 0.1181 3.0000 -90.0000 1.0000 0.1771 4.0000 -120.0000 0 0.2066 5.0000 -132.0000 0 0.1859 6.0000 -120.0000 0 0.1268 7.0000 -78.0000 1.0000 0.0634 8.0000 0 1.0000 0.0219 9.0000 120.0000 1.0000 0.0047 10.0000 288.0000 0 0.0005 [Z,PZ] = csort(G,PX); % Sorting and consolidating to obtain disp([Z;PZ]') % the distribution for Z = g(X) -132.0000 0.1859 -120.0000 0.3334 -90.0000 0.1771 -78.0000 0.0634 -48.0000 0.1181 0 0.0832 48.0000 0.0245 78.0000 0.0000 90.0000 0.0074 120.0000 0.0064 132.0000 0.0003 288.0000 0.0005 P1 = (G<-120)*PX ' % Further calculation using G, PX P1 = 0.1859 p1 = (Z<-120)*PZ' % Alternate using Z, PZ p1 = 0.1859
Example 10.1.12
\(X = 10 I_A + 18 I_B + 10 I_C\) with \(\{A, B, C\}\) independent and \(P =\) [<IP_ADDRESS>].
We calculate the distribution for \(X\), then determine the distribution for
\(Z = X^{1/2} - X + 50\)
c = [10 18 10 0]; pm = minprob(0.1*[6 3 5]); canonic Enter row vector of coefficients c Enter row vector of minterm probabilities pm Use row matrices X and PX for calculations Call for XDBN to view the distribution disp(XDBN) 0 0.1400 10.0000 0.3500 18.0000 0.0600 20.0000 0.2100 28.0000 0.1500 38.0000 0.0900 G = sqrt(X) - X + 50; % Formation of G matrix [Z,PZ] = csort(G,PX); % Sorts distinct values of g(X) disp([Z;PZ]') % consolidates probabilities 18.1644 0.0900 27.2915 0.1500 34.4721 0.2100 36.2426 0.0600 43.1623 0.3500 50.0000 0.1400 M = (Z < 20)|(Z >= 40) % Direct use of Z distribution M = 1 0 0 0 1 1 PZM = M*PZ' PZM = 0.5800
Remark . Note that with the m-function csort, we may name the output as desired.
Example 10.1.13: Continuation of example 10.1.12, above.
H = 2*X.^2 - 3*X + 1; [W,PW] = csort(H,PX) W = 1 171<PHONE_NUMBER> 2775 PW = 0.1400 0.3500 0.0600 0.2100 0.1500 0.0900
Example 10.1.14: A discrete approximation
Suppose \(X\) has density function \(f_X(t) = \dfrac{1}{2} (3t^2 + 2t)\) for \(0 \le t \le 1\). Then \(F_X (t) = \dfrac{1}{2} (t^3 + t^2)\). Let \(Z = X^{1/2}\). We may use the approximation m-procedure tappr to obtain an approximate discrete distribution. Then we work with the approximating random variable as a simple random variable. Suppose we want \(P(Z \le 0.8)\). Now \(Z \le 0.8\) iff \(X \le 0.8^2 = 0.64\). The desired probability may be calculated to be
\(P(Z \le 0.8) = F_X (0.64) = (0.64^3 + 0.64^2)/2 = 0.3359\)
Using the approximation procedure, we have
tappr Enter matrix [a b] of x-range endpoints [0 1] Enter number of x approximation points 200 Enter density as a function of t (3*t.^2 + 2*t)/2 Use row matrices X and PX as in the simple case G = X.^(1/2); M = G <= 0.8; PM = M*PX' PM = 0.3359 % Agrees quite closely with the theoretical


1.E: Chapter One (Exercises)
Is each primarily engaged in rhetoric or primarily engaged in reasoning?
A. We’re going to invade that country, and we’re going to win. The enemies of democracy will find no quarter on the face of the Earth.
B. We’re going to invade that country, and here’s why: the threat the current government poses to the American people is clear and present and furthermore the massacres taking place through the military are just short of genocide. We have, I think, a moral duty to intervene to prevent the further loss of life—American or Foreign.
C. My client is innocent of these trumped up charges and furthermore is the victim of a relentless campaign of defamation motivated by the most cynical and dare I say racist motives this country has seen in a prosecution. Any decent person can see that though my client is no angel, she is nevertheless no murderer.
D. My client is innocent of the charges brought before you in this case. I’ll lay out why in what follows. She has an air tight alibi: she was at her brother’s funeral at the time of the murder. She has no motive: she was taking piano lessons from the victim at the time, but the victim’s logs show that they had a good and healthy rapport. They were even friends on Facebook and those records indicate a friendly relationship.
Is each of the following a proposition or a non-proposition? (remember that propositions are sometimes false)
A. Don’t eat my food when I’ve clearly labeled it.
B. Please try to be more considerate.
C. The moon is made of cheese.
D. The moon is the most awesome thing in the universe.
E. The Betelgeuse Nebula is approximately 2 light years from Earth.
F. How can you act that way when you know it hurts her feelings?
G. Nobody knows what lives on the dark side of the moon.
H. Stop talking about the moon.
I. The dark side of the moon isn’t really dark.
J. Pink Floyd is one of the greatest musical groups of all time.
K. Yaaaaah! Pink Floyd!!!!!
L. For real, though, please stop talking about the moon.
Break down each complex proposition into its component simple propositions. The words ‘and’, ‘but’, ‘or’, ‘if...then...’, and ‘if and only if’ tell us (at least at this introductory stage) we’re dealing with a complex proposition and also tell us what the simple propositions are.
A. Y’all don’t know nothing, but at least I know something that’ll help us get out of this situation.
B. I’ve always been a self-reliant person, and I don’t generally like to travel together, but if you can get me to Chicago, then I’d be mighty grateful.
C. If I had a cow, then I would have all the milk I needed.
D. Either Valeria knew what she was doing or she was completely ignorant of the effects that her actions were likely to have.
E. Robots are either sentient or they are just machines.
F. An ear of corn is either used for animal feed or for applications in human cuisine.
G. Samir, Raj, and Asia were late to school today.
Identify whether each is an argument or inference, or whether it is instead a set of disconnected statements, an explanation of why something happened the way it did , an account or story, or something else that doesn’t have an inferential connection in it.
A. She only cheated on the exam because her financial aid depended on it.
B. We know she cheated on her exam because the instructor caught her stealing glances at her neighbor’s test and found notes hidden in the sleeve of her jacket.
C. He only cheated on his boyfriend because his boyfriend was neglecting him.
D. Rabies is an often-deadly viral infection that causes hydrophobia or the inability to drink water. There are treatments available, but they must be administered quickly.
E.  No one has ever been to the moon, so we don’t know that it isn’t made of cheese.
G. “Boy when you walk by every night talking sweet and looking fine, I get kinda hectic inside.” – Mariah Carey
H. “I never sleep, ‘cause sleep is the cousin of death” – Nas
I. No one should ever sleep, because sleep is the cousin of death
J. Every time I walk by here, you’re sitting around doing nothing.  So was Bobby Kennedy and his brother John. Dr. King and Malcolm were killed around the same time.
L. “Hey Jude. Don’t make it bad. Take a sad song and make it better.” – Paul McCartney
M. “(you shouldn’t) carry the weight of the world on your shoulders, for you know that it's a fool who plays it cool.” (Hey Jude again)
N. We need to reverse our course when it comes to greenhouse gas emissions. If we don’t, the Earth will soon become inhospitable to human life. We must act now.
O. As Joni Mitchell said, we shouldn’t pave paradise and put up a parking lot. To do so would be to make the all-too-common mistake of failing to recognize what we’ve got until it is gone.
For each proposition you find, identify the premises and the conclusions based on the indicators. If there are no indicators for a proposition, try to figure out if it’s a conclusion or a premise.
A. Since we can’t go out until the restaurant opens, and given that we were trying to save money, we should just hang out here until closer to dinner.
B. Why should we save the wetlands, you ask? Simple: because wetlands protect us during storms as they slow water flows and surges.
C. As there has never been a storm of such strength in the Florida Keys, one must conclude that there is not likely to be a storm of such strength in the Florida Keys in the years to come.
D. Nearly every student in the class scored less than 70% on the exam. This entails that the exam was too difficult to accurately assess student learning and therefore, since exams are supposed to accurately assess student learning, the exam must be revised and readministered.
E. Insofar as the losing candidate received a majority of the popular votes, and given that we live in a democracy, we may conclude that the election went against the basic principles of our country.
For each, determine which factual claims are being made and which inferential claims are being made. There will likely be a list of factual claims and then one (often implied) inferential claim.
A. The Affordable Care Act is in a death spiral. Premiums are getting higher, and as premiums get higher, the people will stop purchasing policies, and if the people stop purchasing policies, then the insurance companies will pull out of the exchanges, and if that happens, then the whole system collapses.
B. College isn't designed around the goal of producing good plumbers and electricians and welders. Furthermore, college is expensive and college is time-consuming. So, we shouldn't expect everyone to go to college.
C. We should pass the Affordable Care Act. There is an epidemic of chronically-ill citizens without health insurance due to their pre-existing conditions and many citizens simply can’t afford health insurance.
D. No one wants to be put in the position where they are faced with a deadly intruder without the proper means to protect theirself and their family. Gun laws make it probable that someone will end up in that situation. Therefore, we can’t enact gun control legislation.
For each, supply a charitable interpretation of the argument or position. Give a sort of argument for why someone might believe this or accept this argument. Even if you don’t believe the conclusion, we still need to be able to give the strongest version of the argument or position we’re engaging with.
A. I think the Earth is only about 5,600 years old. The fossil records are inconclusive because carbon dating makes a lot of questionable assumptions.
B. We should keep spending more on the military every year. China is only becoming more of a threat.
C. Borders are not necessary. The concept of a closed border came about fairly recently in human history and having hard borders only creates violence and a coercive market for human trafficking.
D. Policing is inherently unjust. Communities in our society should be able to regulate themselves.
E. We need to build a colony on the Moon!
F. Aliens have certainly visited us in secret.
G. Astrology is real, my personality matches my horoscope more often than not.


11.11: Short Ride in a Fast Machine
As you read this page on Short Ride in a Fast Machine , our playlist example of John Adams’s music, please pay attention both to the ways in which the piece exemplifies the principles of minimalism and the ways in which it expands on those principles (note the references to “post-minimalism”). One of the main elements in this piece is rhythm, in particular the repeated beat played by the woodblock. Note in that section that Adams plays other rhythmic patterns against that beat that shift the listener’s sense of pulse.
Introduction
John Adams completed Short Ride in a Fast Machine in 1986. He applies the description “fanfare for orchestra” to this work and to the earlier Tromba Lontana (1985). The former is also known as Fanfare for Great Woods because it was commissioned for the Great Woods Festival of the Pittsburgh Symphony Orchestra. As a commentary on the title Adams inquires, “You know how it is when someone asks you to ride in a terrific sports car, and then you wish you hadn’t?” This work is an iconic example of Adams’s postminimal style, which is utilized in other works like Phrygian Gates , Shaker Loops , and Nixon in China . This style derives from minimalism as defined by the works of Steve Reich, Terry Riley, andPhilip Glass, although it proceeds to “make use of minimalist techniques in more dramatic settings.”
Rhythmic Devices
In terms of rhythm, this work follows in the main precepts of minimalism, which focus on repeated material, generally in the form of ostinati. There is also a strong sense of pulse, which Adams heavily enforces in Short Ride in a Fast Machine in his scoring of the wood block. Adams claims that “I need to experience that fundamental tick” in his work. Throughout the course of the work, Adams experiments with the idea of rhythmic dissonance as material begins to appear, initially in the trumpets, and gravitates the listener to a new sense of pulse. As shown below, the manifestation of rhythmic dissonance is akin to Adams’s method of creating harmonic dissonance as he adds conflicting rhythms to disrupt the metronomic stability of the wood block. Adams himself admits that he seeks to “enrich the experience of perceiving the way that time is divided” within his works. Later in the work, (see Example 4) Adams introduces a simple polyrhythm as a means of initiating a new section that contrasts the rhythmic dissonance of the first section.
Contributors and Attributions
- Authored by : Elliott Jones. Provided by : Santa Ana College. Located at : http://www.sac.edu . License : CC BY: Attribution
- Short Ride in a Fast Machine. Provided by : Wikipedia. Located at : http://en.Wikipedia.org/wiki/Short_Ride_in_a_Fast_Machine . License : CC BY-SA: Attribution-ShareAlike


1.4: Having Grit
What is grit ? According to the Merriam- Webster Dictionary, grit is “firmness of mind or spirit; unyielding courage in the face of hardship or danger.” If we have grit, we believe we can achieve our goals, even when we’re faced with negative situations. Dr. Angela Duckworth of the University of Pennsylvania has researched this concept of grit with students, business leaders and all types of people (Preister, 2015).
Persevering with “Grit” by Angela Duckworth
Leaving a high-flying job in consulting, Angela Lee Duckworth took a job teaching math to seventh graders in a New York public school. She quickly realized that IQ wasn’t the only thing separating the successful students from those who struggled. Here, she explains her theory of “grit” as a predictor of success (Priester, 2015).
Please click the following link to watch Angela Lee Duckworth’s TED talk, “Grit: The Power of Passion and Perseverance Duckworth The Power of Passion and Perservence TED Talk [www.ted.com]
As powerful as the concept of grit is, there are some limitations. If a person is in an unhealthy environment surrounded by only negative role models and doesn’t see or have any positive influences, grit may not be enough to inspire that person. We do need to see some evidence that our efforts can move us forward, and that there can be a way to leave a negative environment or situation. For this reason, we need to have an internal locus of control and a growth mindset to believe in our abilities to achieve greater goals, and then we draw upon our grit to enable us to do the hard work necessary to move forward.
After watching the videos about fixed/growth mindset and grit, please do some self-examination. Where is your mindset now on the fixed-growth continuum? Do you have grit? Please reflect on how you can apply a “growth mindset” and “grit” to your career and life-planning journey.


1.5: Practice - Text as Process and Outcome
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Practice is another conceptual theme which underpins the studies in this volume. Theoretical approaches to practice (e.g. Bourdieu 1977; Foucault 1979; Giddens 1979; 1984) have been brought to bear on the study of archaeological data for more than three decades (e.g. David and Kramer 2001; Dobres 2000). A dominant concern among these studies has been with technology and charting innovation, change, and continuity. Particular emphasis has been placed on agency, identity, and the body, but in keeping with traditional disciplinary divisions, writing has been largely omitted from this discourse. The recognition engendered by a material practice perspective — that the act of writing and its material products are fundamentally technological — makes it incumbent upon archaeologists to study the marks of inscription in the same way that lithic, ceramic or other types of data are examined.
Similar to analyses of these archaeological data types (Schlanger 1996; Tite 2008), it follows that explanatory frameworks developed for studies of mark-making should also incorporate theories of practice. Etienne Wenger’s concept of “communities of practice”, with its emphasis on learning, and participation and reification (1998: 58–62), offers ways for exploring writing on the levels of both individual and collective practice. Practices are reified, or not, depending on accumulations of individual participation. Reification in everyday life may remain abstract in its manifestation, such as the practice of taking a tea break at an appointed time or shaking hands upon meeting, but reification also shapes experience and meaning in more materially enduring ways. The computer and printing technologies used to produce this volume constitute the nature of writing and reify a particular view of it materially, in contrast to many of the writing practices addressed in the contributions themselves. The concept of “communities of practice” draws on Anthony Giddens’ notion of “structuration” — the negotiation of the relationship between individual agency and social structures through situated practice. This concept of agency as constituted by, and constituting of, social structure ensures a framework for understanding practice that is neither over-individualising nor over-generalising (cf. Gardner 2004: 2–4 with e.g. Barrett 2001: 149; Hodder 2000: 25).
While a concept of agency that is set in relation to social structure can be fruitful for explaining how individuals choose to act and participate (or not) in writing cultures (see Piquette 2013), archaeological theory is also well-equipped to provide new explanatory frameworks for addressing writing in the context of bodily practice. One direction in which engagement with material practice leads us is a concern for the senses, through which human beings experience the material world. The broader spectrum of human sensory experience of past materialities has been investigated within archaeology since the early 1990s and has become more prominent in recent years (Fahlander and Kjellström 2010; Skeates 2010), albeit with limited concern for past writing. The emergence of Visual Cultural Studies during the late 1980s as its own discipline, and the field of Image Studies as well (Mitchell 2002: 178), represents an important move to treat imagery and its materiality from a more multisensory perspective (Jay 2002: 88; despite the visual bias implied in its name), but here too writing has been sidelined. Perhaps some insight into why certain barriers persist for work across some disciplinary boundaries is required. Marquard Smith (2008: 1–2) makes an interesting observation with regard to publication in his discipline, Visual Cultural Studies, which parallels our experience in bringing this volume to press. It is commonplace to encounter numerous books with ‘visual’ and ‘culture’ in the title in university libraries, bookshops or online booksellers, but where they are shelved or how they are otherwise categorised ranges widely. From Art History, Aesthetics and Anthropology to Critical Theory or Sociology, no one is quite sure where to put visual culture or where to find it. The present volume seemed to present a similar classificatory conundrum (and thus marketing difficulties according to one publisher we approached). The ontological challenge presented by the notion of writing as object, and an object that is embedded within the full spectrum of human sensory experience, presents an interesting paradox. If one pauses to survey one’s surroundings, graphical culture of all sorts is clearly embedded in the material world. In the present day we cope easily with the interweaving of writing and associated image types in day-to-day life. Whether we are checking text messages on a phone, flicking through a magazine, licking a stamp, struggling to unfurl a newspaper on a crowded bus, or reading this very text as part of a paper-based or e-book, it is easy to see how these material contexts and sensory experiences beyond the visual are important to writing-related practices and meanings. Yet, as long as we fail to develop an epistemological infrastructure which supports investigation of these complexities, we cannot develop an understanding of the wider networks which constituted past written meaning or properly evaluate its cultural significance. Likewise, archaeological thought on decision-making processes, choice and intentionality also stands to contribute to research on the selection of writing materials, and the choices past people made for how to write, read, view or otherwise engage with written surfaces.
However we understand material practice in general, in any given case study we need to ask both who were the practitioners and how they practised. Here we come up against another set of problematic terms — literacy, reading and writing — on which there is a substantial literature. In the more linguistically oriented studies devoted to the subject of literacy there is a strong emphasis on ‘reading’ and ‘writing,’ understood very much in present day terms (see Collins and Blot 2003 for an overview). Archaeologists and ancient historians have devoted much time to discussion of the extent of literacy in any given society (by which they usually mean the number of people who could read and write, rather than what is indicated by these terms; see, for instance, Harris’s seminal work Ancient Literacy (Harris 1989) and the responses of a number of other scholars (Beard et al. 1991). However, the kind of approach adopted in this volume requires the reconsideration of definitions of both ‘writer’ and ‘reader’ and also to consider a wider range of practitioners than can be encompassed in these terms, for instance the people who made the artefacts, who may well have been different from the people who wrote on them.
When thinking about ‘writers’ we need to be explicit about whether we mean the people who wielded the pen, stylus, brush or chisel, or those who composed the message. These may have been the same people, but equally may not have been, especially where materials were used that required complex technologies and specialist artisans. We also need to consider the role of people commissioning an inscription who might not themselves have been able to write or read. For instance, the production of a bronze tablet to be put up in a public place, as known from the Roman world, might involve four different types of maker: a member of the political or religious establishment to commission the work, a literate bureaucrat to compose the text, a bronze smith to fashion the tablet, and probably a different bronze worker to chisel the letters. Of these people, only the bureaucrat had to be literate, in the sense of understanding the sense of the text. The person who produced the actual writing (whom one might think reasonable to label the ‘writer’) might have been copying a prototype and have had little understanding of what the text meant. Maureen Carroll (2009: 47) mentions a splendid example of this, the Roman stone funerary inscription from Annaba that reads hic iacet corpus pueri nominandi (here lies the body of the boy... insert name): the letter cutter had failed to notice that he was meant to insert a specific name!
‘Readers’ are equally difficult to define. We might identify fully literate (in the modern sense) readers, who could understand texts completely; we might also consider those who could perhaps read a little, but could not decipher a text in detail. There would be others who could not read at all but who ‘consumed’ writing through oral performance by others. Or those who did not even do this but who viewed the texts and knew they were important in some way. And who were the readers of hidden inscriptions (those on the inside of sealed tombs or even built into the construction itself)? If the intended viewers were dead people or supernatural beings, in what sense were they ‘readers’?


12.4: Scottish Traditional - “Scotland the Brave”
Our previous example, “The Stars and Stripes Forever,” might just as well have been discussed in Chapter 9 as an example of music embodying national identity. Not only is it the official National March of the United States, but it has come to be associated with patriotic events, especially Independence Day. Our next example would be even more at home in Chapter 9: Both the tune—“Scotland the Brave”1—and the musical tradition—Scottish Highland bagpiping—are intimately connected with Scottish history and identity. This is, of course, hardly surprising. When we talk about marches, we are often talking about military tradition, and it is nearly impossible to disentangle military history and practice from national identity and pride.
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“Scotland the Brave” 1. Performance: Pipes and Drums of the Gordon Highlanders (2008) |
Scottish Highland Bagpipes
Scottish Highland bagpipes are certainly a military instrument. The first written account of bagpipes being carried into battle dates from 1549, when the piercing sound of bagpipes was found to carry across the battlefield even better than that of a trumpet. Pipers were regularly attached to combat regiments, and they could be heard on both sides of the famous Jacobite rising of 1745, when Charles Edward Stuart sought to regain the British throne. In this conflict, bagpipes served not only to urge troops into battle but to mark the identity of the combatants: the Jacobites carried Scottish Highland bagpipes, while the British troops carried Lowland bagpipes (a related instrument that has fallen out of use).
Pipers continue to serve in the British military into the present day, although their role is now only ceremonial and they are no longer deployed in combat. The last appearance of pipers on the front line took place during a 1967 uprising in the British protectorate of South Arabia (now part of Yemen). Pipers also led assaults during both of the World Wars, but the practice was largely abolished due to the high casualty rate.
There is no question that the Scottish Highland bagpipes are an outdoor instrument. They are extremely loud. Scottish bagpipes seem to have come into use during the 1200s, although it is unknown whether they were adapted from a Roman instrument or developed independently. The basic principle of any bagpipe is the same: An air reservoir (usually a bag) is squeezed so as to force air through a series of pipes, thereby producing sound. Most of the pipes are tuned to a fixed pitch, but one is equipped with holes that allow the piper to sound specific notes. While bagpipes can be found around the world, the Scottish instrument and associated tradition has certainly gained the greatest prominence. Today, pipe bands can be found in almost all former British colonies and throughout the United States. In Scottish Highland bagpipes, the air reservoir is filled when the piper blows into a long air pipe . Although the piper must regularly refill the reservoir, this action has no impact on the sound of the instrument, which is continuous and always at the same volume. This is because the sound is produced by air from the bag—which is kept under a constant pressure—flowing out through the pipes. The fixed-pitched pipes are three in number, and they all play the same note, although in two different octaves. The other pipe—termed the chanter —allows the piper to play nine distinct pitches spanning just over an octave.
The fact that the melodic capabilities of the Highland pipes is so limited means that performers must either play repertoire suited to their instrument or adapt other melodies by changing some of the notes. For example, pipers frequently play the melody from Dvořák’s Symphony No. 9 “From the New World” (Chapter 9), which is usually retitled “Goin’ Home.”2 However, the Highland pipes do not have a pitch that is required to correctly perform the middle section of the melody. Instead, pipers simply play the closest pitch that is available on the instrument, which happens to be one half step lower than the pitch Dvořák had intended.
2. “Goin’ Home” is a popular selection for Highland pipers.
It is typical for pipers to borrow tunes from other repertoires. In fact, the two tunes most frequently played on Scottish Highland bagpipes—“Amazing Grace” and “Scotland the Brave”—began life and became popular independent of the piping tradition. “Amazing Grace,” discussed in Chapter 11, is a Christian hymn. Although the lyrics were written by an English preacher, they were paired with their melody in the United States, which is also where the hymn became popular. In a sense, therefore, “Amazing Grace”—like “Goin’ Home”—is a musical import.
Scotland the Brave
“Scotland the Brave,” on the other hand, seems to have been written in Scotland in the 1890s, although it was initially played on other instruments. The tune first appeared in the 1895 Gesto Collection of Highland Music , in which it was titled “Scotland Forever” and labelled as a “trumpet march.” Around 1911, it was included in a collection of pipe music published by the Boys’ Brigade Bands of Glasgow—an indication that Scottish pipers were playing it. The tune, however, did not become famous until 1951, when Cliff Hanley added a patriotic text so that it could be sung in a theatrical production. “Scotland the Brave” quickly became a hit, and in 2006 placed second in a public poll to select an unofficial national anthem for Scotland. “Scotland the Brave” bears all of the typical characteristics of a march. It is in duple time, to facilitate the tramping of feet. It is in a bright major mode (although the tuning of pipes is not always equivalent to that of a piano). The melody is rhythmically straightforward: There is a note on every strong beat, meaning that the melody will coordinate with the steps of marchers, and the notes that fall between the beats are placed sparsely and predictably. After all, a complex melody would be lost in the noise of an outdoor march, and would confuse the regular rhythm of the movement. It might even prove catastrophic to coordination. The melody also features uneven, or dotted , rhythms, which have long been associated with royalty and pomp.
“Scotland the Brave” is also packed with repetition. It is in a rounded binary form , meaning that its two parts both end with the same melodic phrase. That phrase also constitutes the first half of the first part, such that the tune can be diagrammed as [A A’] [B A’]. In a single turn through the tune, therefore, we hear the A phrase three times—and the B phrase is closely related, bearing the same rhythmic characteristics. Although this form is common in folk music of the British Isles, the melody of “Scotland the Brave” is particularly effective. Both the A and B phrases include the lowest and highest notes of the tune, although they reverse the melodic contour. The A phrase ascends immediately from the low octave to the high and then back down, while the B phrase begins in the high octave and plummets to the low—twice. The result is dizzyingly triumphant.
When we hear pipers play “Scotland the Brave,” we hear more than just the notes of the melody as it might be sung or played on the trumpet. That is because pipers deploy a variety of idiomatic melodic ornaments that are uniquely available on their instruments. These sometimes sound like glitches or hiccups, but they are in fact specific sequences of between one and four pitches that are quickly and precisely added to the melody using the finger holes on the chanter. Each ornament has a name and is considered appropriate for a specific type of music, and pipers in large bands perform ornaments in perfect unison. In “Scotland the Brave,” ornaments help to emphasize the march rhythm. After all, the Scottish Highland bagpipes can only produce sound in a constant, uninterrupted stream. Ornaments break up the sound and introduce rhythmic excitement.
We hear several other instruments on our recording. The pipers are aided by a line of drummers, who keep the pulse steady while also emphasizing the rhythmic contour of the melody. The second time through the melody the pipers are joined by the instruments of a marching band, including brass and flutes. These instruments play a simpler, unornamented version of the tune. We can detect subtle clashes when the pitch produced by the bagpipes, which are tuned according to a unique system, disagrees with that produced by the other instruments.


5.2: Tidal Curvature Versus Curvature Caused by Local Sources
A further complication is the need to distinguish tidal curvature from curvature caused by local sources. Figure \(\PageIndex{1}\) shows Comet Shoemaker-Levy, broken up into a string of fragments by Jupiter’s tidal forces shortly before its spectacular impact with the planet in 1994. Immediately after each fracture, the newly separated chunks had almost zero velocity relative to one another, so once the comet finished breaking up, the fragments’ world-lines were a sheaf of nearly parallel lines separated by spatial distances of only 1 km. These initially parallel geodesics then diverged, eventually fanning out to span millions of kilometers.
If initially parallel lines lose their parallelism, that is clearly an indication of intrinsic curvature. We call it a measure of sectional curvature , because the loss of parallelism occurs within a particular plane, in this case the \((t, x)\) plane represented by Figure \(\PageIndex{2}\).
But this curvature was not caused by a local source lurking in among the fragments. It was caused by a distant source: Jupiter. We therefore see that the mere presence of sectional curvature is not enough to demonstrate the existence of local sources. Even the sign of the sectional curvature is not a reliable indication. Although this example showed a divergence of initially parallel geodesics, referred to as a negative curvature, it is also possible for tidal forces exerted by distant masses to create positive curvature. For example, the ocean tides on earth oscillate both above and below mean sea level, Figure Figure \(\PageIndex{3}\).
As an example that really would indicate the presence of a local source, we could release a cloud of test masses at rest in a spherical shell around the earth, and allow them to drop, Figure \(\PageIndex{4}\). We would then have positive and equal sectional curvature in the \(t − x\), \(t − y\), and \(t − z\) planes. Such an observation cannot be due to a distant mass. It demonstrates an over-all contraction of the volume of an initially parallel sheaf of geodesics, which can never be induced by tidal forces. The earth’s oceans, for example, do not change their total volume due to the tides, and this would be true even if the oceans were a gas rather than an incompressible fluid. It is a unique property of \(\frac{1}{r^{2}}\) forces such as gravity that they conserve volume in this way; this is essentially a restatement of Gauss’s law in a vacuum.


3.2: Motions of Objects
There are several basic motions of bodies. Rotation is the spinning of an object around its axis. Earth rotates once approximately every 24 hours, whereas Jupiter rotates once approximately every 10 hours.
At any moment, half of the Earth is illuminated by the Sun, because of the fact that most of our light comes from the Sun. As the Earth rotates from west to east, your location moves from the dark (night) hemisphere into the illuminated (day) hemisphere and back again. The diurnal (daily) motion of the stars, the Sun, and the Moon is a consequence of Earth’s rotation.
Another motion is Revolution; that is, one object orbiting a second. Earth revolves around the Sun approximately every 365. 24 Earth Days (1 Earth year), and Jupiter revolves around the Sun approximately every 11. 86 Earth Years (1 Jovian year). As the Earth revolves or orbits around the Sun, the nighttime side of the Earth (or any other body orbiting the Sun) gradually turns toward different parts of the night sky.